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THEOBY    OF    FUNCTIONS 


OF   A 


COMPLEX    VARIABLE. 


Itonlion:    C.  J.  CLAY  AND  SONS, 

CAMBEIDGE   UNIVEESITY  PEESS   WAEEHOUSE, 

AVE   MAEIA  LANE. 


CAMBEIDGE  :  DEIGHTON,  BELL,  AKD  CO. 

LEIPZIG  :  F.  A.  BROCKHAUS. 
NEW  YORK:   MACMILLAN  AND  CO. 


THEOEY    OF    FUNCTIONS 


OF    A 


COMPLEX   VARIABLE 


BY 


A.    R    FORSYTE,    So.D.,    F.RS., 

FELLOW   OF   TRINITY   COLLEGE,   CAMBRIDGE. 


CAMBEIDGE: 
AT    THE    UNIVERSITY    PRESS. 

1893 

All  rights  reserved.  . 


Mtth.  U.  01. 


PRINTED    BY    C.    J.    CLAY,    M.A.    AND    SONS, 
AT    THE    UNIVERSITY    PRESS. 


PEEFACE. 

AMONG  the  many  advances  in  the  progress  of  mathematical 
XlL  science  during  the  last  forty  years,  not  the  least  remarkable 
are  those  in  the  theory  of  functions.  The  contributions  that  are 
still  being  made  to  it  testify  to  its  vitality  :  all  the  evidence  points 
to  the  continuance  of  its  growth.  And,  indeed,  this  need  cause  no 
surprise.  Few  subjects  can  boast  such  varied  processes,  based 
upon  methods  so  distinct  from  one  another  as  are  those  originated 
by  Cauchy,  by  Weierstrass,  and  by  Biemann.  Each  of  these 
methods  is  sufficient  in  itself  to  provide  a  complete  development ; 
combined,  they  exhibit  an  unusual  wealth  of  ideas  and  furnish 
unsurpassed  resources  in  attacking  new  problems. 

It  is  difficult  to  keep  pace  with  the  rapid  growth  of  the 
literature  which  is  due  to  the  activity  of  mathematicians, 
especially  of  continental  mathematicians :  and  there  is,  in  con 
sequence,  sufficient  reason  for  considering  that  some  marshalling 
of  the  main  results  is  at  least  desirable  and  is,  perhaps,  necessary. 
Not  that  there  is  any  dearth  of  treatises  in  French  and  in 
German  :  but,  for  the  most  part,  they  either  expound  the  pro 
cesses  based  upon  some  single  method  or  they  deal  with  the 
discussion  of  some  particular  branch  of  the  theory. 


814033 


PREFACE 


The  present  treatise  is  an  attempt  to  give  a  consecutive 
account  of  what  may  fairly  be  deemed  the  principal  branches  of 
the  whole  subject.  It  may  be  that  the  next  few  years  will  see 
additions  as  important  as  those  of  the  last  few  years  :  this  account 
would  then  be  insufficient  for  its  purpose,  notwithstanding  the 
breadth  of  range  over  which  it  may  seem  at  present  to  extend. 
My  hope  is  that  the  book,  so  far  as  it  goes,  may  assist  mathe 
maticians,  by  lessening  the  labour  of  acquiring  a  proper  knowledge 
of  the  subject,  and  by  indicating  the  main  lines,  on  which  recent 
progress  has  been  achieved. 

No  apology  is  offered  for  the  size  of  the  book.  Indeed,  if 
there  were  to  be  an  apology,  it  would  rather  be  on  the  ground 
of  the  too  brief  treatment  of  some  portions  and  the  omissions 
of  others.  The  detail  in  the  exposition  of  the  elements  of  several 
important  branches  has  prevented  a  completeness  of  treatment 
of  those  branches  :  but  this  fulness  of  initial  explanations  is 
deliberate,  my  opinion  being  that  students  will  thereby  become 
better  qualified  to  read  the  great  classical  memoirs,  by  the  study 
of  which  effective  progress  can  best  be  made.  And  limitations  of 
space  have  compelled  me  to  exclude  some  branches  which  other 
wise  would  have  found  a  place.  Thus  the  theory  of  functions  of 
a  real  variable  is  left  undiscussed  :  happily,  the  treatises  of  Dini, 
Stolz,  Tannery  and  Chrystal  are  sufficient  to  supply  the  omission. 
Again,  the  theory  of  functions  of  more  than  one  complex  variable 
receives  only  a  passing  mention  ;  but  in  this  case,  as  in  most 
cases,  where  the  consideration  is  brief,  references  are  given 
which  will  enable  the  student  to  follow  the  development  to 
such  extent  as  he  may  desire.  Limitation  in  one  other  direction 
has  been  imposed  :  the  treatise  aims  at  dealing  with  the  general 
theory  of  functions  and  it  does  not  profess  to  deal  with  special 
classes  of  functions.  I  have  not  hesitated  to  use  examples  of 
special  classes  :  but  they  are  used  merely  as  illustrations  of  the 
general  theory,  and  references  are  given  to  other  treatises  for 
the  detailed  exposition  of  their  properties. 


PREFACE  Vll 

The  general  method  which  is  adopted  is  not  limited  so  that 
it  may  conform  to  any  single  one  of  the  three  principal  inde 
pendent  methods,  due  to  Cauchy,  to  Weierstrass  and  to  Biemann 
respectively :  where  it  has  been  convenient  to  do  so,  I  have 
combined  ideas  and  processes  derived  from  different  methods. 

The  book  may  be  considered  as  composed  of  five  parts. 

The  first  part,  consisting  of  Chapters  I — VII,  contains  the 
theory  of  uniform  functions  :  the  discussion  is  based  upon  power- 
series,  initially  connected  with  Cauchy's  theorems  in  integration, 
and  the  properties  established  are  chiefly  those  which  are  con 
tained  in  the  memoirs  of  Weierstrass  and  Mittag-Leffler. 

The  second  part,  consisting  of  Chapters  VIII — XIII,  contains 
the  theory  of  multiform  functions,  and  of  uniform  periodic 
functions  which  are  derived  through  the  inversion  of  integrals 
of  algebraic  functions.  The  method  adopted  in  this  part  is 
Cauchy's,  as  used  by  Briot  and  Bouquet  in  their  three  memoirs 
and  in  their  treatise  on  elliptic  functions  :  it  is  the  method  that 
has  been  followed  by  Hermite  and  others  to  obtain  the  properties 
of  various  kinds  of  periodic  functions.  A  chapter  has  been 
devoted  to  the  proof  of  Weierstrass's  results  relating  to  functions 
that  possess  an  addition-theorem. 

The  third  part,  consisting  of  Chapters  XIV — XVIII,  contains 
the  development  of  the  theory  of  functions  according  to  the 
method  initiated  by  Biemann  in  his  memoirs.  The  proof  which 
is  given  of  the  existence-theorem  is  substantially  due  to  Schwarz ; 
in  the  rest  of  this  part  of  the  book,  I  have  derived  great  assist 
ance  from  Neumann's  treatise  on  Abelian  functions,  from  Fricke's 
treatise  on  Klein's  theory  of  modular  functions,  and  from  many 
memoirs  by  Klein. 

The  fourth  part,  consisting  of  Chapters  XIX  and  XX,  treats 
of  conformal  representation.  The  fundamental  theorem,  as  to  the 
possibility  of  the  conformal  representation  of  surfaces  upon  one 
another,  is  derived  from  the  existence-theorem  :  it  is  a  curious  fact 
that  the  actual  solution,  which  has  been  proved  to  exist  in  general, 
F.  b 


Vlll  PREFACE 

has  been  obtained  only  for  cases  in  which  there  is  distinct 
limitation. 

The  fifth  part,  consisting  of  Chapters  XXI  and  XXII,  contains 
an  introduction  to  the  theory  of  Fuchsian  or  automorphic  functions, 
based  upon  the  researches  of  Poincare  and  Klein  :  the  discussion  is 
restricted  to  the  elements  of  this  newly-developed  theory. 

The  arrangement  of  the  subject-matter,  as  indicated  in  this 
abstract  of  the  contents,  has  been  adopted  as  being  the  most 
convenient  for  the  continuous  exposition  of  the  theory.  But  the 
arrangement  does  not  provide  an  order  best  adapted  to  one  who  is 
reading  the  subject  for  the  first  time.  I  have  therefore  ventured 
to  prefix  to  the  Table  of  Contents  a  selection  of  Chapters  that 
will  probably  form  a  more  suitable  introduction  to  the  subject  for 
such  a  reader ;  the  remaining  Chapters  can  then  be  taken  in  an 
order  determined  by  the  branch  of  the  subject  which  he  wishes 
to  follow  out. 

In  the  course  of  the  preparation  of  this  book,  I  have  consulted 
many  treatises  and  memoirs.  References  to  them,  both  general 
and  particular,  are  freely  made  :  without  making  precise  reserva 
tions  as  to  independent  contributions  of  my  own,  I  wish  in  this 
place  to  make  a  comprehensive  acknowledgement  of  my  obligations 
to  such  works.  A  number  of  examples  occur  in  the  book  :  most  of 
them  are  extracted  from  memoirs,  which  do  not  lie  close  to  the 
direct  line  of  development  of  the  general  theory  but  contain 
results  that  provide  interesting  special  illustrations.  My  inten 
tion  has  been  to  give  the  author's  name  in  every  case  where  a 
result  has  been  extracted  from  a  memoir  :  any  omission  to  do  so 
is  due  to  inadvertence. 

Substantial  as  has  been  the  aid  provided  by  the  treatises  and 
memoirs  to  which  reference  has  just  been  made,  the  completion  of 
the  book  in  the  correction  of  the  proof-sheets  has  been  rendered 
easier  to  me  by  the  unstinted  and  untiring  help  rendered  by 
two  friends.  To  Mr  William  Burnside,  M.A.,  formerly  Fellow  of 


PREFACE 


Pembroke  College,  Cambridge,  and  now  Professor  of  Mathematics 
at  the  Royal  Naval  College,  Greenwich,  I  am  under  a  deep  debt 
of  gratitude  :  he  has  used  his  great  knowledge  of  the  subject  in 
the  most  generous  manner,  making  suggestions  and  criticisms  that 
have  enabled  me  to  correct  errors  and  to  improve  the  book  in 
many  respects.  Mr  H.  M.  Taylor,  M.  A.,  Fellow  of  Trinity  College, 
Cambridge,  has  read  the  proofs  with  great  care  :  the  kind  assist 
ance  that  he  has  given  me  in  this  way  has  proved  of  substantial 
service  and  usefulness  in  correcting  the  sheets.  I  desire  to 
recognise  most  gratefully  my  sense  of  the  value  of  the  work  which 
these  gentlemen  have  done. 

It  is  but  just  on  my  part  to  state  that  the  willing  and  active 
co-operation  of  the  Staff  of  the  University  Press  during  the  pro 
gress  of  printing  has  done  much  to  lighten  my  labour. 

It  is,  perhaps,  too  ambitious  to  hope  that,  on  ground  which 
is  relatively  new  to  English  mathematics,  there  will  be  freedom 
from  error  or  obscurity  and  that  the  mode  of  presentation  in  this 
treatise  will  command  general  approbation.  In  any  case,  my  aim 
has  been  to  produce  a  book  that  will  assist  mathematicians  in 
acquiring  a  knowledge  of  the  theory  of  functions  :  in  proportion 
as  it  may  prove  of  real  service  to  them,  will  be  my  reward. 

A.  R.  FORSYTE. 


TRINITY  COLLEGE,  CAMBRIDGE. 
25  February,  1893. 


CONTENTS. 


The  following  course  is  recommended,  in  the  order  specified,  to  those  who  are 
reading  the  subject  for  the  first  time  :  The  theory  of  uniform  functions,  Chapters 
I— V ;  Conformal  representation,  Chapter  XIX  ;  Multiform  functions  and  uniform 
periodic  functions,  Chapters  VIII— XI ;  Riemanris  surfaces,  and  Riemann's  theory 
of  algebraic  functions  and  their  integrals,  Chapters  XIV— XVI,  XVIII. 


CHAPTER   I. 

GENERAL   INTRODUCTION. 

§§ 

PAGE 

1—3.      The  complex  variable  and  the  representation  of  its  variation  by  points 

in  a  plane , 

4.  Neumann's  representation  by  points  on  a  sphere      ...  4 

5.  Properties  of  functions  assumed  known      ...  Q 
6,  7.      The  idea  of  complex  functionality  adopted,   with  the  conditions  neces 
sary  and  sufficient  to  ensure  functional  dependence  ...  6 

8.  Riemann's  definition  of  functionality          ...  g 

9.  A   functional    relation    between   two    complex   variables   establishes   the 

geometrical  property  of  conformal  representation  of  their  planes  .  10 

10,  11.  Relations  between  the  real  and  the  imaginary  parts  of  a  function  of  z  11 
12,  13.  Definitions  and  illustrations  of  the  terms  monogenic,  uniform,  multiform, 

branch,  branch-point,  holomorphic,  zero,  pole,  meromorphic  .         .         .  14 


CHAPTER   II. 

INTEGRATION   OF   UNIFORM   FUNCTIONS. 

14,  15.     Definition  of  an  integral  with  complex  variables  ;   inferences  .        .        .  '       18 
16.        Proof    of   the    lemma     I  I  (^  -  £  \  dxdy=\(pdx -\-qdy),  under  assigned 

I     \  fll'1  (it I  I  J     •*  J.       •/  f '  O 


conditions 21 


CONTENTS 


§§  PAGE 

17,  18.     The  integral  \f(z)dz  round    any  simple   curve  is  zero,   when  f(z)   is 

Cz 

holomorphic   within   the   curve;    and    I    /(*)<&    is    a    holomorphic 

J  a 

function  when  the  path  of  integration  lies  within  the  curve     .         .         23 
19.        The   path   of  integration   of  a  holomorphic  function    can    be   deformed 

without  changing  the  value  of  the  integral         .....         26 

20—22.    The  integral  =—  .  I  '—  '-  dz,   round   a   curve    enclosing   a,   is   /(a)    when 
27rt  J  z  —  a 

f(z)  is  a  holomorphic  function  within  the  curve;  and  the  integral 

J_  [    /(*)      dz  is   —  ,—^.     Superior  limit  for  the  modulus  of 
27rt  J(z-a)n  +  1  n\     dan 

the  nth  derivative  of  /(a)  in  terms  of  the  modulus  of  /(a)     .         .         27 

23.  The  path  of  integration  of  a  meromorphic  function  cannot  be  deformed 

across  a  pole  without  changing  the  value  of  the  integral.         .         .         34 

24.  The  integral  of  any  function  (i)  round  a  very  small  circle,  (ii)  round  a 

very  large  circle,  (iii)  round  a  circle  which  encloses  all  its  infinities 

and  all  its  branch-points      .........         35 

25.  Special  examples  ............         •• 


CHAPTER   III. 

EXPANSIONS   OF   FUNCTIONS   IN    SERIES   OF   POWERS. 

26,  27.  Cauchy's  expansion  of  a  function  in  positive  powers  of  z  -  a ;  with  re 
marks  and  inferences 43 

28—30.    Laurent's  expansion   of  a  function  in  positive   and   negative   powers  of 

z  -  a ;   with  corollary 47 

31.        Application  of  Cauchy's  expansion  to  the  derivatives  of  a  function         .         51 

32,  33.  Definition  of  an  ordinary  point  of  a  function,  of  the  domain  of  an 
ordinary  point,  of  an  accidental  singularity,  and  of  an  essential 
singularity  .......•••••  52 

34,  35.     Continuation   of    a   function   by   means   of   elements   over    its   region    of 

continuity 54 

36.        Schwarz's   theorem  on  symmetric   continuation  across  the  axis   of  real 

quantities 57 


CHAPTER   IV. 

UNIFORM   FUNCTIONS,  PARTICULARLY  THOSE   WITHOUT   ESSENTIAL 
SINGULARITIES. 

37.        A  function,   constant   over  a  continuous   series   of    points,   is   constant 

everywhere  in  its  region  of  continuity 59 

38,  39.     The  multiplicity  of  a  zero,  which  is  an  ordinary  point,  is  finite;   and 

a  multiple  zero  of  a  function  is  a  zero  of  its  first  derivative  .         .         61 


CONTENTS  Xlll 

§§  PAGE 

40.        A  function,  that  is  not  a  constant,  must  have  infinite  values         .         .  63 

41,  42.     Form  of  a  function  near  an  accidental  singularity 64 

43,  44.     Poles  of  a  function  are  poles  of  its  derivatives          .....  66 
45,  46.     A   function,  which   has  infinity  for  its  only  pole   and  has  no   essential 

singularity,  is  an  algebraical  polynomial      ......  69 

47.  Integral  algebraical  and  integral  transcendental  functions          ...  70 

48.  A  function,  all  the  singularities  of  which  are  accidental,  is  an  algebraical 

meromorphic  function  ..........  71 


CHAPTER   V. 

TRANSCENDENTAL   INTEGRAL   UNIFORM   FUNCTIONS. 

49,  50.     Construction  of  a  transcendental  integral  function  with  assigned   zeros 
a1?  a2,  a3,  ...,  when  an  integer  s  can  be  found  such  that  2|an|~8 

is  a  converging  series 74 

51.        Weierstrass's  construction  of  a  function  with  any  assigned  zeros    .         .         77 
52,  53.     The  most    general    form    of   function   with   assigned   zeros   and  having 

its  single  essential  singularity  at  0=00        .         .         .         .         .         .         80 

54.        Functions  with  the  singly-infinite  system  of  zeros  given  by  ;Z  =  TO<B,  for 

integral  values  of  m 82 

55 — 57.     Weierstrass's  o--function  with  the  doubly-infinite  system  of  zeros  given 

by  z=ma>  +  m'a>,  for  integral  values  of  TO  and  of  TO' .         .         .         .         84 
58.        A  function  cannot  exist  with  a  triply-infinite  arithmetical  system  of  zeros         88 

59,  60.     Class  (genre)  of  a  function 89 

61.        Laguerre's  criterion  of  the  class  of  a  function 91 


CHAPTER  VI. 

FUNCTIONS  WITH   A   LIMITED   NUMBER   OF   ESSENTIAL   SINGULARITIES. 

62.  Indefiniteness  of  value  of  a  function  at  an  essential  singularity       .         .         94 

63.  A  function  is  of  the  form  O  {  — =- )  +  P  (z  —  6)  in  the  vicinity  of  an  essen- 

\»— o/ 

tial  singularity  at  b,  a  point  in  the  finite  part  of  the  plane  .  .  96 
64,  65.  Expression  of  a  function  with  n  essential  singularities  as  a  sum  of  n 

functions,  each  with  only  one  essential  singularity  ....  99 
66,  67.  Product-expression  of  a  function  with  n  essential  singularities  and  no 

zeros  or  accidental  singularities 101 

68 — 71.  Product-expression  of  a  function  with  n  essential  singularities  and  with 

assigned   zeros  and   assigned  accidental  singularities ;   with   a  note 

on  the  region  of  continuity  of  such  a  function  .         .         .  .104 


xiv  CONTENTS 


CHAPTER   VII. 

FUNCTIONS   WITH   UNLIMITED   ESSENTIAL   SINGULARITIES, 
AND   EXPANSION    IN   SERIES   OF   FUNCTIONS. 

§§  1>AGE 

72.  Mittag-Leffler's    theorem    on    functions  with   unlimited   essential   singu 

larities,  distributed  over  the  whole  plane 112 

73.  Construction  of  subsidiary  functions,  to  be  terms  of  an  infinite  sum     .       113 
74_76.    Weierstrass's  proof  of  Mittag-Leffler's  theorem,  with  the  generalisation 

of  the  form  of  the  theorem 114 

77,  78.     Mittag-Leffler's  theorem   on  functions   with    unlimited  essential    singu 
larities,  distributed  over  a  finite  circle 117 

79.  Expression  of  a  given  function  in  Mittag-Leffler's  form    ....  123 

80.  General  remarks  on  infinite  series,  whether  of  powers  or  of  functions    .  126 

81.  A  series  of  powers,  in  a  region  of  continuity,  represents  one  and  only 

one  function  ;  it  cannot  be  continued  beyond  a  natural  limit .         .       128 

82.  Also  a  series  of  functions  :   but  its  region  of  continuity  may  consist  of 

distinct  parts 129 

83.  A  series  of  functions  does  not  necessarily  possess  a  derivative  at  points 

on  the  boundary  of  any  one  of  the  distinct  portions  of  its  region 

of  continuity          ........•••       133 

84.  A  series  of  functions  may  represent  different  functions  in  distinct  parts 

of  its  region  of  continuity  ;  Tannery's  series 136 

85.  Construction  of  a  function  which  represents  different  assigned  functions 

in  distinct  assigned  parts  of  the  plane        .         .         .         .         .         .138 

86.  Functions  with  a  line  of  essential  singularity 139 

87.  Functions  with  an  area  of  essential  singularity  or  lacunary  spaces         .       141 

88.  Arrangement  of  singularities  of  functions  into  classes  and  species  .         .       146 


CHAPTER   VIII. 

MULTIFORM   FUNCTIONS. 

89.  Branch-points  and  branches  of  functions 149 

90.  Branches   obtained   by  continuation:   path   of  variation  of  independent 

variable  between   two   points   can   be  deformed  without  affecting  a 
branch  of  a  function  if  it  be  not  made  to  cross  a  branch-point       .       150 

91.  If  the  path  be  deformed  across  a  branch-point  which  affects  the  branch, 

then  the  branch  is  changed I55 

92.  The  interchange  of  branches  for  circuits- round  a  branch-point  is  cyclical  156 

93.  Analytical  form  of  a  function  near  a  branch-point 157 

94.  Branch-points  of  a  function  defined  by  an  algebraical  equation  in  their 

relation  to  the  branches  :  definition  of  algebraic  function          .         .       161 
95,  96.     Infinities  of  an  algebraic  function 163 


CONTENTS  xv 

PAGE 


97.  Determination  of  the  branch-points  of  an  algebraic  function,  and  of  the 

cyclical  systems  of  the  branches  of  the  function        ...  168 

98.  Special  case,  when  the  branch-points  are  simple  :  their  number      .         .       174 

99.  A  function,  with  n  branches  and  a  limited  number  of  branch-points  and 

singularities,  is  a  root  of  an  algebraical  equation  of  degree  n.         .       175 


CHAPTER   IX. 

PERIODS   OF   DEFINITE   INTEGRALS,   AND   PERIODIC    FUNCTIONS 
IN   GENERAL. 

100.  Conditions   under  which   the  path   of  variation  of  the   integral  of  a 

multiform  function  can  be  deformed  without   changing  the  value 

of  the  integral   .......  J§Q 

101.  Integral   of  a  multiform   function   round  a   small   curve   enclosing   a 

branch-point       .......  183 

102.  Indefinite  integrals  of  uniform  functions  with  accidental  singularities  ; 

fdz      f  dz 

j  i  '       2  .....    •    .....    184 


103.  Hermite's  method  of  obtaining  the  multiplicity  in  value  of  an  integral; 

sections  in  the  plane,  made  to  avoid  the  multiplicity     .         .         .185 

104.  Examples  of  indefinite  integrals  of  multiform  functions  ;    \wdz  round 

any  loop,  the  general  value  of  J(l  -  22)  ~  *  dz,  of  J{1  -  z2)  (1  -  k^}}  ~  *  dz, 

and  of  5{(z-el)(z-e2)(g-es)}-*dz  .......       189 

105.  Graphical    representation   of    simply-periodic    and   of    doubly-periodic 

functions  .......  198 

106.  The  ratio  of  the  periods  of  a  uniform  doubly-periodic  function  is  not 

real     .............       200 

107,  108.     Triply-periodic  uniform  functions  of  a  single  variable  do  not  exist     .       202 

109.  Construction   of  a  fundamental  parallelogram  for  a  uniform   doubly- 

periodic  function         .......  205 

110.  An  integral,  with  more  periods  than  two,  can  be  made  to  assume  any 

value  by  a  modification   of  the   path  of  integration  between  the 
limits          ........  208 


CHAPTER   X. 

SIMPLY-PERIODIC   AND    DOUBLY-PERIODIC   FUNCTIONS. 

2rrzi 

111.  Simply-periodic  functions,  and  the  transformation  Z=e  w  .         ;'    •    .       211 

112.  Fourier's  series  and  simply-periodic  functions 213 

113,  114.     Properties  of  simply-periodic  functions  without  essential  singularities 

in  the  finite  part  of  the  plane 214 

115.  Uniform   doubly-periodic   functions,  without  essential  singularities   in 

the  finite  part  of  the  plane 218 

116.  Properties  of  uniform  doubly-periodic  functions 219 


CONTENTS 

§§ 

117.         The  zeros  and  the  singularities  of  the  derivative  of  a  doubly-periodic 

function  of  the  second  order    .        . 231 

118,  119.     Kelations  between  homoperiodic  functions      .         .        ...         .         •       233 


CHAPTER  XL 

DOUBLY-PERIODIC   FUNCTIONS   OF   THE   SECOND   ORDER. 

120    121.     Formation   of  an   uneven   function  with   two   distinct   irreducible   in 
finities;   its  addition-theorem 243 

122,  123.     Properties  of  Weierstrass's  o-function     .        .        .        .                 •         •       247 
124.         Introduction  of  f  (2)  and  of   Q(z) 250 

125,  126.     Periodicity  of  the  function  #>  (z),  with  a  single   irreducible   infinity  of 

degree  two;  the  differential  equation  satisfied  by  the  function  #>  (2)       251 

127.  Pseudo-periodicity  of  f(«) •       255 

128.  Construction   of  a  doubly-periodic  function   in   terms  of  f  (z)  and  its 

derivatives • .  -     .         •         •  256 

129.  On  the  relation  qw'-  j/eo  =  ±%iri 25>7 

130.  Pseudo-periodicity  of  a  (z) •  259 

131.  Construction  of  a  doubly-periodic  function  as  a  product  of  o-functions ; 

with  examples 259 

132.  On   derivatives   of   periodic  functions   with   regard   to   the   invariants 

#2  and  £3 *        '         '       lfK 

133 135.    Formation  of  an  even  function  of  either  class 266 


CHAPTER  XII. 

PSEUDO-PERIODIC   FUNCTIONS. 

136.         Three  kinds  of  pseudo-periodic  functions,  with  the  characteristic  equa 
tions  273 

137,  138.     Hermite's  and  Mittag-Leffler's  expressions  for  doubly-periodic  functions 

of  the  second  kind 275 

139.         The  zeros  and  the  infinities  of  a  secondary  function    .         .     -  .         .  280 

140,  141.     Solution  of  Lamp's  differential  equation 281 

142.  The  zeros  and  the  infinities  of  a  tertiary  function         ....  286 

143.  Product-expression  for  a  tertiary  function 287 

144—146.    Two  classes  of  tertiary  functions ;   Appell's  expressions  for  a  function 

of  each  class  as  a  sum  of  elements 288 

147.  Expansion  in  trigonometrical  series 293 

148.  Examples  of  other  classes  of  pseudo-periodic  functions          .         .         .       295 


CONTENTS  Xvii 

CHAPTER  XIII. 

FUNCTIONS   POSSESSING   AN   ALGEBRAICAL   ADDITION-THEOREM. 
§§  PAGE 

149.  Definition  of  an  algebraical  addition-theorem 297 

150.  A    function    defined   by    an   algebraical    equation,    the    coefficients    of 

which  are  algebraical  functions,  or  simply-periodic  functions,  or 
doubly-periodic  functions,  has  an  algebraical  addition-theorem  .  297 

151 — 154.  A  function  possessing  an  algebraical  addition-theorem  is  either 
algebraical,  simply-periodic  or  doubly-periodic,  having  in  each 
instance  only  a  finite  number  of  values  for  an  argument  .  .  300 

155,  156.  A  function  with  an  algebraical  addition-theorem  can  be  defined  by  a 
differential  equation  of  the  first  order,  into  which  the  independent 
variable  does  not  explicitly  enter 309 

CHAPTER   XIV. 

CONNECTIVITY   OF   SURFACES. 

157—159.  Definitions  of  connection,  simple  connection,  multiple  connection,  cross 
cut,  loop-cut  .  .  .  .'.•-.,  312 

160.  Relations  between  cross-cuts  and  connectivity 315 

161.  Relations  between  loop-cuts  and  connectivity 320 

162.  Effect  of  a  slit .321 

163,  164.     Relations  between  boundaries  and  connectivity 322 

165.  Lhuilier's    theorem    on    the    division    of    a    connected    surface    into 

curvilinear  polygons 325 

166.  Definitions  of  circuit,  reducible,  irreducible,  simple,  multiple,  compound, 

reconcileable 327 

167,  168.     Properties  of  a  complete   system   of  irreducible   simple  circuits  on  a 

surface,  in  its  relation  to  the  connectivity 328 

169.  Deformation  of  surfaces 332 

170.  Conditions  of  equivalence  for  representation  of  the  variable         .         .  333 


CHAPTER   XV. 

RIEMANN'S  SURFACES. 

171.  Character  and  general  description  of  a  Riemann's  surface    .     -  ..  •.       336 

172.  Riemann's  surface  associated  with  an  algebraical  equation   .         .  .338 

173.  Sheets  of  the  surface  are  connected  along  lines,  called  branch-lines  .       338 

174.  Properties  of  branch-lines 340 

175,  176.     Formation  of  system  of  branch-lines  for  a  surface ;   with  examples  .       341 

177.         Spherical  form  of  Riemann's  surface       .         .         .  34(5 


XV111  CONTENTS 

§§  PAGE 

178.  The  connectivity  of  a  Eiemann's  surface 347 

179.  Irreducible   circuits  :    examples    of  resolution   of    Riemann's    surfaces 

into  surfaces  that  are  simply  connected 350 

180,  181.     General  resolution  of  a  Riemann's  surface 353 

182.  A  Riemann's  %-sheeted  surface  when  all  the  branch-points  are  simple  355 

183,  184.     On  loops,  and  their  deformation 356 

185.         Simple  cycles  of  Clebsch  and  Gordan 359 

186 — 189.  Canonical  form   of  Riemann's  surface  when  all  the  branch -points  are 

simple,  deduced  from  theorems  of  Luroth  and  Clebsch.         .         .  361 

190.  Deformation  of  the  surface 365 

191.  Remark  on  uniform  algebraical  transformations 367 


CHAPTER   XVI. 

ALGEBRAIC    FUNCTIONS   AND   THEIR    INTEGRALS. 

192.         Two  subjects  of  investigation 368 

193,  194.     Determination  of  the  most  general  uniform  function  of  position   on  a 

Riemann's  surface      ..........  369 

195.         Preliminary  lemmas  in  integration  on  a  Riemann's  surface  .         .         .  372 

196,  197.     Moduli  of  periodicity  for  cross-cuts  in  the  resolved  surface  .         .         .  373 

198.  The  number  of  linearly  independent  moduli  of  periodicity  is  equal  to 

the  number  of  cross-cuts,  which  are  necessary  for  the  resolution 

of  the  surface  into  one  that  is  simply  connected   ....       378 

199.  Periodic  functions  on  a  Riemann's  surface,  with  examples    .         .         .       379 

200.  Integral    of  the    most    general   uniform    function    of    position    on     a 

Riemann's  surface      .         .         .         ; 387 

201.  Integrals,  everywhere  finite  on  the   surface,  connected  with  the  equa 

tion  w*=S(z) 388 

202 — 204.  Infinities  of  integrals  on  the  surface  connected  with  the  algebraical 
equation  f  (w,  z)  =  0,  when  the  equation  is  geometrically  interpret- 
able  as  the  equation  of  a  (generalised)  curve  of  the  nth  order  .  388 

205,  206.  Integrals  of  the  first  kind  connected  with/(w,  z)  =  0,  Demg  functions 
that  are  everywhere  finite :  the  number  of  such  integrals,  linearly 
independent  of  one  another :  they  are  multiform  functions  .  .  394 

207,  208.     Integrals  of  the  second  kind  connected  with  f  (w,  z)  =  0,  being  func 
tions  that  have  only  algebraical  infinities;   elementary  integral  of 
the  second  kind         ..........       396 

209.         Integrals  of  the  third  kind  connected  with/(w,  z)  =  0,  being  functions 

that  have  logarithmic  infinities 400 

210,  211.     An  integral  of  the  third  kind  cannot  have  less  than  two  logarithmic 

infinities ;    elementary  integral  of  the  third  kind     ....       401 


CONTENTS 


CHAPTER  XVII. 

SCHWARZ'S   PROOF   OF   THE   EXISTENCE-THEOREM. 

§§  PAGE 
212,  213.     Existence   of  functions   on   a  Riemann's   surface;   initial  limitation  of 

the  problem  to  the  real  parts  u  of  the  functions  .         ...      .         .  405 

214.  Conditions  to  which  u,  the  potential  function,  is  subject       .         .         .  407 

215.  Methods  of  proof :   summary  of  Schwarz's  investigation         .         .         .  408 
216 — 220.    The  potential-function  u  is  uniquely  determined  for  a  circle  by  the  gene 
ral  conditions  and  by  the  assignment  of  finite  boundary  values   .  410 

221.  Also  for  any  plane  area,  on  which  the  area  of  a  circle  can  be  con- 

formally  represented 423 

222.  Also  for  any  plane  area  which  can  be  obtained  by  a  topological  com 

bination  of  areas,  having  a  common  part   and   each  conformally 
representable  on  the  area  of  a  circle 425 

223.  Also  for  any  area   on   a   Riemann's   surface  in  which  a  branch-point 

occurs ;   and  for  any  simply  connected  surface        ....       428 
224 — 227.    Real  functions  exist  on  a   Riemann's   surface,  everywhere   finite,  and 

having  arbitrarily  assigned  real  moduli  of  periodicity     .         .         .       430 

228.  And  the  number  of  the  linearly  independent  real  functions  thus  ob 

tained  is  2p       ...........       434 

229.  Real    functions    exist  with    assigned    infinities    on    the    surface    and 

assigned   real  moduli  of  periodicity.     Classes   of  functions  of  the 
complex  variable  proved  to  exist  on  the  Riemann's  surface  .         .       435 


CHAPTER   XVIII. 

APPLICATIONS   OF   THE   EXISTENCE-THEOREM. 

230.         Three  special  kinds  of  functions  on  a  Riemann's  surface     .         .         .       437 
231 — 233.    Relations  between  moduli  of  functions  of  the  first  kind  and  those  of 

functions  of  the  second  kind 439 

234.  The  number  of  linearly  independent  functions  of  the  first  kind  on  a 

Riemann's  surface  of  connectivity  2/;  +  l  is  p 

235.  Normal  functions  of  the  first  kind ;   properties  of  their  moduli    . 

236.  Normal  elementary  functions  of  the  second  kind :    their  moduli   . 
237,  238.     Normal  elementary  functions  of  the  third  kind :   their  moduli :   inter 
change  of  arguments  and  parametric  points 449 

239.  The  inversion-problem  for  functions  of  the  first'  kind    ....       453 

240.  Algebraical  functions  on  a  Riemann's  surface  without  infinities  at  the 

branch-points  but  only  at  isolated  ordinary  points  on  the  surface : 
Riemann-Roch's  theorem :  the  smallest  number  of  singularities 
that  such  functions  may  possess  .  .  .  .  .  .  .457 

241.  A  class  of  algebraic  functions  infinite  only  at  branch-points         .         .       460 

242.  Fundamental  equation  associated  with  an  assigned  Riemann's  surface       462 


XX  CONTENTS 

§§  PAGE 

243.         Appell's  factorial  functions  on  a  Riemann's  surface :    their  multipliers 

at  the  cross-cuts 464 

244,  245.  Expression  for  a  factorial  function  with  assigned  zeros  and  assigned 
infinities;  relations  between  zeros  and  infinities  of  a  factorial 
function  .  .  ...  .  .  .'  .  •  •  •  466 

246.         Functions  defined  by  differential  equations  of  the  form  /  ( w,  -y- )  =  0        470 

\       **&) 

247 — 249.    Conditions  that  the  function  should  be  a  uniform  function  of  z.         .       471 
250,  251.     Classes  of  uniform  functions  that  can  be  so  defined,  with  criteria  of 

discrimination    .         .         •    ,    •         •         •         •         •         •         •         •       476 

(dw\s 
~T~  )  =/  (w)        ....       482 
az  j 


CHAPTER   XIX. 

CONFORMAL   REPRESENTATION  :     INTRODUCTORY. 

253.  A  relation  between  complex  variables  is  the  most  general  relation  that 

secures  conformal  similarity  between  two  surfaces ....       491 

254.  One  of  the  surfaces  for  conformal  representation  may,  without  loss  of 

generality,  be  taken  to  be  a  plane 495 

255,  256.     Application  to  surfaces  of  revolution ;   in  particular,  to   a   sphere,  so 

as  to  obtain  maps     ..........       496 

257.  Some   examples   of  conformal  representation   of  plane  areas,    in   par 

ticular,  of  areas  that  can  be  conformally  represented  on  the  area 

of  a  circle 501 

258.  Linear    homographic    transformations    (or    substitutions)    of    the    form 

w  = ,:   their  fundamental  properties 512 

cz  +  d 

259.  Parabolic,  elliptic,  hyperbolic  and  loxodromic  substitutions    .         .         .       517 

260.  An   elliptic   substitution   is  either  periodic   or  infinitesimal :  substitu 

tions  of  the  other  classes  are  neither  periodic  nor  infinitesimal    .       521 

261.  A   linear  substitution   can  be  regarded  geometrically  as  the  result   of 

an  even  number  of  successive  inversions  of  a  point  with  regard 

to  circles    .  .........       523 


CHAPTER   XX. 

CONFORMAL  REPRESENTATION  :  GENERAL  THEORY. 

262.  Riemann's  theorem  on  the  conformal  representation  of  a  given  area 

upon  the  area  of  a  circle  with  unique  correspondence  .  .  .  525 

263,  264.  Proof  of  Riemann's  theorem  :  how  far  the  functional  equation  is 

algebraically  determinate 526 

265,  266.  The  method  of  Beltrami  and  Cayley  for  the  construction  of  the 

functional  equation  for  an  analytical  curve 530 


CONTENTS  XXI 

§§  PAGE 

267,  268.     Conformal   representation    of  a   convex    rectilinear  polygon    upon  the 

half-plane  of  the  variable 537 

269.  The  triangle,  and  the  quadrilateral,  conformally  represented         .         .       543 

270.  A  convex  curve,  as  a  limiting  case  of  a  polygon  ....       548 
271,  272.     Conformal  representation  of  a  convex  figure,  bounded  by  circular  arcs  : 

the  functional    relation    is    connected   with   a    linear    differential 
equation  of  the  second  order    ........       549 

273.         Conformal  representation  of  a  crescent  .......       554 

274 — 276.    Conformal  representation  of  a  triangle,  bounded  by  circular  arcs         .       555 
277 — 279.    Relation  between  the  triangle,  bounded  by  circles,  and  the  stereographic 

projection  of  regular  solids  inscribed  in  a  sphere  ....       563 

280.         On  families  of  plane  algebraical  curves,  determined  as  potential-curves 
by   a   single  parameter  u  +  vi  :   the    forms   of    functional   relation 

),  which  give  rise  to  such  curves  ....       575 


CHAPTER   XXI. 

GROUPS   OF   LINEAR   SUBSTITUTIONS. 

281.  The  algebra  of  group-symbols 582 

282.  Groups,   which   are   considered,   are   discontinuous   and  have   a  finite 

number  of  fundamental  substitutions 584 

283,  284.  Anharmonic  group  :  group  for  the  modular-functions,  and  division  of 

the  plane  of  the  variable  to  represent  the  group  ....  586 
285,  286.  Fuchsian  groups  :  division  of  plane  into  convex  curvilinear  polygons : 

polygon  of  reference 591 

287.  Cycles  of  angular  points  in  a  curvilinear  polygon  ....  595 

288,  289.  Character  of  the  division  of  the  plane  :  example  ....  599 

290.  Fuchsian  groups  which  conserve  a  fundamental  circle  .        .         .  602 

291.  Essential   singularities  of  a  group,  and  of  the  automorphic  functions 

determined  by  the  group          ........       605 

292,  293.     Families  of  groups  :    and  their  class 606 

294.  Kleinian  groups  :   the  generalised  equations  connecting  two  points  in 

space "...        .        .        .609 

295.  Division  of  plane  and  division  of    space,  in  connection  with  Kleinian 

groups 613 

296.  Example  of  improperly  discontinuous  group 615 


CHAPTER   XXII. 

AUTOMORPHIC    FUNCTIONS. 

297.  Definition  of  automorphic  functions         .         .         .         .        .    '-    .        .619 

298.  Examples  of  functions,  automorphic  for  finite  discrete  groups  of  sub 

stitutions   620 

299.  Cayley's  analytical  relation  between  stereographic  projections  of  posi 

tions  of  a  point  on  a  rotated  sphere 620 


XX11  CONTENTS 

§§  PAGE 

300.         Polyhedral  groups ;   in  particular,  the  dihedral  group,  and  the  tetra- 

hedral  group 623 

301,  302.     The  tetrahedral  functions,  and  the  dihedral  functions  .         .         .       628 

303.  Special    illustrations    of    infinite    discrete    groups,    from    the    elliptic 

modular-functions 633 

304.  Division  of  the  plane,  and  properties   of  the  fundamental  polygon  of 

reference,  for  any  infinite  discrete  group  that  conserves  a  funda 
mental  circle      ...........       637 

305,  306.     Construction   of   Thetafuchsian  functions,   pseudo-automorphic   for  an 

infinite  group  of  substitutions 641 

307.  Relations  between   the  number  of  irreducible  zeros   and  the  number 

of  irreducible  poles  of  a  pseudo-automorphic  function,  constructed 

with  a  rational  algebraical  meromorphic  function  as  element        .       645 

308.  Construction  of  automorphic  functions   .......       650 

309.  The  number  of  irreducible  points,  for  which  an  automorphic  function 

acquires  an  assigned  value,  is  independent  of  the  value         .         .       651 

310.  Algebraical   relations   between    functions,    automorphic    for   a   group  : 

application  of  Riemanu's  theory  of  functions  ....       653 

311.  Connection    between    automorphic    functions    and    linear    differential 

equations  ;    with  illustrations  from  elliptic  modular-functions        .       654 


GLOSSARY  OF  TECHNICAL  TERMS  .  .  .  .  ;  .  .  '  ,  .  659 
INDEX  OF  AUTHORS  QUOTED  .  .  .  .  -  .  .  .  .  .  662 
GENERAL  INDEX  664 


CHAPTER  I. 

GENERAL  INTRODUCTION. 

1.  ALGEBRAICAL  operations  are  either  direct  or  inverse.  Without 
entering  into  a  general  discussion  of  the  nature  of  irrational  and  of  imaginary 
quantities,  it  will  be  sufficient  to  point  out  that  direct  algebraical  operations 
on  numbers  that  are  positive  and  integral  lead  to  numbers  of  the  same 
character;  and  that  inverse  algebraical  operations  on  numbers  that  are 
positive  and  integral  lead  to  numbers,  which  may  be  negative  or  fractional 
or  irrational,  or  to  numbers  which  may  not  even  fall  within  the  class  of  real 
quantities.  The  simplest  case  of  occurrence  of  a  quantity,  which  is  not 
real,  is  that  which  arises  when  the  square  root  of  a  negative  quantity  is 
required. 

Combinations  of  the  various  kinds  of  quantities  that  may  occur  are  of 
the  form  x  +  iy,  where  x  and  y  are  real  and  i,  the  non-real  element  of  the 
quantity,  denotes  the  square  root  of  - 1.  It  is  found  that,  when  quantities 
of  this  character  are  subjected  to  algebraical  operations,  they  always  lead  to 
quantities  of  the  same  formal  character;  and  it  is  therefore  inferred  that 
the  most  general  form  of  algebraical  quantity  is  x  +  iy. 

Such  a  quantity  ic  +  iy,  for  brevity  denoted  by  z,  is  usually  called  a 
complex  variable*;  it  therefore  appears  that  the  complex  variable  is  the 
most  general  form  of  algebraical  quantity  which  obeys  the  fundamental  laws 
of  ordinary  algebra. 

2.  The  most  general  complex  variable  is  that,  in  which  the  constituents 
x  and  y  are  independent  of  one  another  and  (being  real  quantities)  are 
separately  capable  of  assuming  all  values  from  -  oo  to  +  oo  ;  thus  a  doubly- 
infinite  variation  is  possible  for  the  variable.  In  the  case  of  a  real  variable, 
it  is  convenient  to  use  the  customary  geometrical  representation  by  measure 
ment  of  distance  along  a  straight  line;  so  also  in  the  case  of  a  complex 


*  The  conjugate  complex,  viz.  x  -  iy,  is  frequently  denoted  by  za. 
F. 


2  GEOMETRICAL   REPRESENTATION   OF  [2. 

variable,  it  is  convenient  to  associate  a  geometrical  representation  with 
the  algebraical  expression ;  and  this  is  the  well-known  representation  of 
the  variable  ac  +  iy  by  means  of  a  point  with  coordinates  x  and  y  referred 
to  rectangular  axes*.  The  complete  variation  of  the  complex  variable  z 
is  represented  by  the  aggregate  of  all  possible  positions  of  the  associated 
point,  which  is  often  called  the  point  z ;  the  special  case  of  real  variables 
being  evidently  included  in  it  because,  when  y  =  0,  the  aggregate  of 
possible  points  is  the  line  which  is  the  range  of  geometrical  variation  of 
the  real  "variable. 

•  The  variation  of  z  is  said  to  be  continuous  when  the  variations  of  x  and  y 
are  contiguous.  Continuous  variation  of  z  between  two  given  values  will 
thus  be  represented  by  continuous  variation  in  the  position  of  the  point  z, 
that  is,  by  a  continuous  curve  (not  necessarily  of  continuous  curvature) 
between  the  points  corresponding  to  the  two  values.  But  since  an  infinite 
number  of  curves  can  be  drawn  between  two  points  in  a  plane,  continuity  of 
line  is  not  sufficient  to  specify  the  variation  of  the  complex  variable ;  and,  in 
order  to  indicate  any  special  mode  of  variation,  it  is  necessary  to  assign, 
either  explicitly  or  implicitly,  some  determinate  law  connecting  the  variations 
of  x  and  y  or,  what  is  the  same  thing,  some  determinate  law  connecting  x 
and  y.  The  analytical  expression  of  this  law  is  the  equation  of  the  curve 
which  represents  the  aggregate  of  values  assumed  by  the  variable  between 
the  two  given  values. 

In  such  a  case  the  variable  is  often  said  to  describe  the  part  of  the  curve 
between  the  two  points.  In  particular,  if  the  variable  resume  its  initial 
value,  the  representative  point  must  return  to  its  initial  position ;  and  then 
the  variable  is  said  to  describe  the  whole  curve -f-. 

When  a  given  closed  curve  is  continuously  described  by  the  variable, 
there  are  two  directions  in  which  the  description  can  take  place.  From 
the  analogy  of  the  description  of  a  straight  line  by  a  point  representing  a 
real  variable,  one  of  these  directions  is  considered  as  positive  and  the  other 

*  This  method  of  geometrical  representation  of  imaginary  quantities,  ordinarily  assigned  to 
Gauss,  was  originally  developed  by  Argand  who,  in  1806,  published  his  "  Essai  sur  une  maniere 
de  representer  les  quantites  imaginaires  dans  les  constructions  geometriques."  This  tract  was 
republished  in  1874  as  a  second  edition  (Gauthier-Villars) ;  an  interesting  preface  is  added 
to  it  by  Hoiiel,  who  gives  an  account  of  the  earlier  history  of  the  publications  associated  with 
the  theory. 

Other  references  to  the  historical  development  are  given  in  Chrystal's  Text-book  of  Algebra, 
vol.  i,  pp.  248,  249;  in  Holzmiiller's  Einfilhrung  in  die  Theorie  der  isogonalen  Venvandschaften 
und  dcr  conformen  Abbildungen,  verbunden  mit  Anwendungen  auf  mathematische  Physik,  pp.  1 — 10, 
21 — 23 ;  in  Schlomilch's  Compendium  der  hoheren  Analysis,  vol.  ii,  p.  38  (note) ;  and  in  Casorati, 
Teorica  delle  funzioni  di  variabili  complesse,  only  one  volume  of  which  was  published.  In  this 
connection,  an  article  by  Cayley  (Quart.  Journ.  of  Math,,  vol.  xxii,  pp.  270 — 308)  may  be 
consulted  with  advantage. 

t  In  these  elementary  explanations,  it  is  unnecessary  to  enter  into  any  discussion  of 
the  effects  caused  by  the  occurrence  of  singularities  in  the  curve. 


2-] 


THE   COMPLEX    VARIABLE 


Fig.  1. 


as  negative.  The  usual  convention  under  which  one  of  the  directions  is 
selected  as  the  positive  direction  depends  upon  the  conception  that  the  curve 
is  the  boundary,  partial  or  complete,  of  some  area ;  under  it,  that  direction  is 
taken  to  be  positive  which  is  such  that  the  bounded  area  lies  to  the  left  of 
the  direction  of  description.  It  is  easy  to  see  that  the  same  direction  is  taken 
to  be  positive  under  an  equivalent  convention 
which  makes  it  related  to  the  normal  drawn 
outwards  from  the  bounded  area  in  the  same 
way  as  the  positive  direction  of  the  axis  of  y 
is  to  the  positive  direction  of  the  axis  of  x 
in  plane  coordinate  geometry. 

Thus  in  the  figure  (fig.  1),  the  positive 
direction  of  description  of  the  outer  curve 
for  the  area  included  by  it  is  DEF;  the 
positive  direction  of  description  of  the  inner 
curve  for  the  area  without  it  (say,  the  area 
excluded  by  it)  is  AGB ;  and  for  the  area 
between  the  curves  the  positive  direction  of  description  of  the  boundary, 
which  consists  of  two  parts,  is  DEF,  ACB. 

3.  Since  the  position  of  a  point  in  a  plane  can  be  determined  by  means 
of  polar  coordinates,  it  is  convenient  in  the  discussion  of  complex  variables 
to  introduce  two  quantities  corresponding  to  polar  coordinates. 

In  the  case  of  the  variable  z,  one  of  these  quantities  is  (#2  +  yn-)l,  the 
positive  sign  being  always  associated  with  it ;  it  is  called  the  modulus*  of 
the  variable  and  it  is  denoted,  sometimes  by  mod.  z,  sometimes  by  \z  . 

The  other  is  0,  the  angular  coordinate  of  the  point  z ;  it  is  called  the 
argument  (and,  less  frequently,  the  amplitude)  of  the  variable.  It  is 
measured  in  the  trigonometrically  positive  sense,  and  is  determined  by 
the  equations 

<K=\Z\  cos  6,     y=  z\  sin#, 

so  that  z=  z\eei.     The  actual  value  depends  upon  the  way  in  which  the 
variable  has  acquired  its  value  ;  when  variation 
of  the  argument  is  considered,  its  initial  value 
is  usually  taken  to  lie  between  0  and  2?r  or,  less 
frequently,  between  -TT  and  +TT. 

As  z  varies  in  position,  the  values  of  \z\ 
and  6  vary.  When  z  has  completed  a  positive 
description  of  a  closed  curve,  the  modulus  of  z 
returns  to  the  initial  value  whether  the  origin  Fig.  2. 


Der  absolute  Metro,;)  is  often  used  by  German  writers. 


1—2 


GREAT   VALUES   OF 


[3. 


be  without,  within  or  on  the  curve.  The  argument  of  z  resumes  its  initial 
value,  if  the  origin  0'  (fig.  2)  be  without  the  curve ;  but,  if  the  origin  0  be 
within  the  curve,  the  value  of  the  argument  is  increased  by  2-rr  when  z 
returns  to  its  initial  position. 

If  the  origin  be  on  the  curve,  the  argument  of  z  undergoes  an  abrupt 
change  by  TT  as  z  passes  through  the  origin ;  and  the  change  is  an  increase 
or  a  decrease  according  as  the  variable  approaches  its  limiting  position  on  the 
curve  from  without  or  from  within.  No  choice  need  be  made  between  these 
alternatives;  for  care  is  always  exercised  to  choose  curves  which  do  not 
introduce  this  element  of  doubt. 

4.  Representation  on  a  plane  is  obviously  more  effective  for  points  at  a 
finite  distance  from  the  origin  than  for  points  at  a  very  great  distance. 

One  method  of  meeting  the  difficulty  of  representing  great  values  is  to 
introduce  a  new  variable  z1  given  by  z'z=\\  the  part  of  the  new  plane  for 
z  which  lies  quite  near  the  origin  corresponds  to  the  part  of  the  old  plane 
for  z  which  is  very  distant.  The  two  planes  combined  give  a  complete 
representation  of  variation  of  the  complex  variable. 

Another  method,  in  many  ways  more  advantageous,  is  as  follows.  Draw 
a  sphere  of  unit  diameter,  touching  the  2-plane  at  the  origin  0  (fig.  3)  on 
the  under  side:  join  a  point  z  in  the  plane  to  0',  the  other  extremity  of 
the  diameter  through  0,  by  a  straight  line  cutting  the  sphere  in  Z. 
Then  Z  is  a  unique  representative  of  z,  that  is,  a  single  point  on  the 
sphere  corresponds  to  a  single  point  on  the  plane  :  and  therefore  the  variable 
can  be  represented  on  the  surface  of  the  sphere.  With  this  mode  of 


Fig.  3. 


representation,  0'  evidently  corresponds  to  an  infinite  value  of  z :  and  points 
at  a  very  great  distance  in  the  2-plane  are  represented  by  points  in  the 
immediate  vicinity  of  0'  on  the  sphere.  The  sphei-e  thus  has  the  advantage 
of  putting  in  evidence  a  part  of  the  surface  jn  which  the  variations  of 


4.]  THE   COMPLEX   VARIABLE  5 

great  values  of  z  can  be  traced*,  and  of  exhibiting  the  uniqueness  of 
z  —  oo  as  a  value  of  the  variable,  a  fact  that  is  obscured  in  the  represen 
tation  on  a  plane. 

The  former  method  of  representation  can  be  deduced  by  means  of  the 
sphere.  At  0'  draw  a  plane  touching  the  sphere :  and  let  the  straight  line 
OZ  cut  this  plane  in  z'.  Then  z  is  a  point  uniquely  determined  by  Z 
and  therefore  uniquely  determined  by  z.  In  this  new  /-plane  take  axes 
parallel  to  the  axes  in  the  2-plane. 

The  points  z  and  /  move  in  the  same  direction  in  space  round  00' 
as  an  axis.  If  we  make  the  upper  side  of  the  2-plane  correspond  to  the 
lower  side  of  the  /-plane,  and  take  the  usual  positive  directions  in  the 
planes,  being  the  positive  trigonometrical  directions  for  a  spectator  looking 
at  the  surface  of  the  plane  in  which  the  description  takes  place,  we  have 
these  directions  indicated  by  the  arrows  at  0  and  at  0'  respectively,  so 
that  the  senses  of  positive  rotations  in  the  two  planes  are  opposite  in 
space.  Now  it  is  evident  from  the  geometry  that  Oz  and  O'z'  are 
parallel ;  hence,  if  0  be  the  argument  of  the  point  z  and  &  that  of  the 
point  z  so  that  6  is  the  angle  from  Ox  to  Oz  and  6'  the  angle  from  O'x' 
to  O'z,  we  have 

6  +  ff  =  ZTT. 

Oz      00' 
Further,  by  similar  triangles,         -^-t  =  ^-f , 

that  is,  Oz .  O'z'  =  OO'2  =  1. 

Now,  if  z  and  z'  be  the  variables,  we  have 

z=0z.eei,     z'=0'z'.effi, 
so  that  zz'=0z.0'z' .e^s'^ 

=  1, 
which  is  the  former  relation. 

The  /-plane  can  therefore  be  taken  as  the  lower  side  of  a  plane  touching 
the  sphere  at  0'  when  the  2-plane  is  the  upper  side  of  a  plane  touching 
it  at  0.  The  part  of  the  2-plane  at  a  very  great  distance  is  represented  on 
the  sphere  by  the  part  in  the  immediate  vicinity  of  0' :  and  this  part  of 
the  sphere  is  represented  on  the  /-plane  by  its  portion  in  the  immediate 
vicinity  of  0',  which  therefore  is  a  space  wherein  the  variations  of  infinitely 
great  values  of  z  can  be  traced. 

But  it  need  hardly  be  pointed  out  that  any  special  method  of  represent 
ation  of  the  variable  is  not  essential  to  the  development  of  the  theory  of 
functions ;  and,  in  particular,  the  foregoing  representation  of  the  variable, 
when  it  has  very  great  values,  merely  provides  a  convenient  method  of 
dealing  with  quantities  that  tend  to  become  infinite  in  magnitude. 

*  This  sphere  is  sometimes  called  Neumann's  sphere;  it  is  used  by  him  for  the  representation 
of  the  complex  variable  throughout  his  treatise  Vorlesungen  uber  Riemann'a  Theorie  der  AlcVschen 
Integrate  (Leipzig,  Teubner,  '2nd  edition,  1884). 


6  CONDITIONS   OF  [5. 

5.  The   simplest    propositions   relating    to   complex   variables   will   be 
assumed  known.     Among  these  are,  the  geometrical  interpretation  of  opera 
tions  such  as  addition,  multiplication,  root-extraction ;  some  of  the  relations 
of   complex  variables  occurring  as  roots  of   algebraical  equations  with  real 
coefficients;   the   elementary  properties   of   functions   of  complex   variables 
which   are  algebraical  and  integral,  or  exponential,   or  circular    functions; 
and  simple  tests  of  convergence  of  infinite  series  and  of  infinite  products*. 

6.  All  ordinary  operations  effected  on  a  complex  variable  lead,  as  already 
remarked,    to    other    complex    variables;    and   any   definite    quantity,    thus 
obtained  by  operations  on  z,  is  necessarily  a  function  of  z. 

But  if  a  complex  variable  w  be  given  as  a  complex  function  of  x 
and  y  without  any  indication  of  its  source,  the  question  as  to  whether 
w  is  or  is  not  a  function  of  z  requires  a  consideration  of  the  general  idea 
of  functionality. 

It  is  convenient  to  postulate  u  +  iv  as  a  form  of  the  complex  variable  w, 
where  u  and  v  are  real.  Since  w  is  initially  unrestricted  in  variation,  we 
may  so  far  regard  the  quantities  u  and  v  as  independent  and  therefore  as 
any  functions  of  x  and  y,  the  elements  involved  in  z.  But  more  explicit 
expressions  for  these  functions  are  neither  assigned  nor  supposed. 

The  earliest  occurrence  of  the  idea  of  functionality  is  in  connection  with 
functions  of  real  variables ;  and  then  it  is  coextensive  with  the  idea  of 
dependence.  Thus,  if  the  value  of  X  depends  on  that  of  x  and  on  no  other 
variable  magnitude,  it  is  customary  to  regard  X  as  a  function  of  x\  and 
there  is  usually  an  implication  that  X  is  derived  from  x  by  some  series  of 
operations^. 

A  detailed  knowledge  of  z  determines  x  and  y  uniquely  ;  hence  the  values 
of  u  and  v  may  be  considered  as  known  and  therefore  also  w.  Thus  the 
value  of  w  is  dependent  on  that  of  z,  and  is  independent  of  the  values 
of  variables  unconnected  with  z;  therefore,  with  the  foregoing  view  of 
functionality,  w  is  a  function  of  z. 

It  is,  however,  equally  consistent  with  that  view  to  regard  w  as  a  complex 
function  of  the  two  independent  elements  from  which  z  is  constituted ;  and 
we  are  then  led  merely  to  the  consideration  of  functions  of  two  real 
independent  variables  with  (possibly)  imaginary  coefficients. 

*  These  and  other  introductory  parts  of  the  subject  are  discussed  in  Chrystal's  Text-book  of 
Algebra  and  in  Hobson's  Treatise  on  Plane  Trigonometry. 

They  are  also  discussed  at  some  length  in  the  recently  published  translation,  by  G.  L. 
Cathcart,  of  Harnack's  Elements  of  the  differential  and  integral  calculus  (Williams  and  Norgate, 
1891),  the  second  and  the  fourth  books  of  which  contain  developments  that  should  be  consulted 
in  special  relation  with  the  first  few  chapters  of  the  present  treatise. 

These  books,  together  with  Neumann's  treatise.cited  in  the  note  on  p.  5,  will  hereafter  be  cited 
by  the  names  of  their  respective  authors. 

t  It  is  not  important  for  the  present  purpose  to  keep  in  view  such  mathematical  expressions 
as  have  intelligible  meanings  only  when  the  independent  variable  is  confined  within  limits. 


6.]  FUNCTIONAL    DEPENDENCE  7 

Both  of  these  aspects  of  the  dependence  of  w  on  z  require  that  z  be 
regarded  as  a  composite  quantity  involving  two  independent  elements  which 
can  be  considered  separately.  Our  purpose,  however,  is  to  regard  z  as  the 
most  general  form  of  algebraical  variable  and  therefore  as  an  irresoluble 
entity  ;  so  that,  as  this  preliminary  requirement  in  regard  to  z  is  unsatisfied, 
neither  of  the  aspects  can  be  adopted. 

7.  Suppose  that  w  is  regarded  as  a  function  of  z  in  the  sense  that  it  can 
be  constructed  by  definite  operations  on  z  regarded  as  an  irresoluble 
magnitude,  the  quantities  u  and  v  arising  subsequently  to  these  operations 
by  the  separation  of  the  real  and  the  imaginary  parts  when  z  is  replaced  by 
x  +  iy.  It  is  thereby  assumed  that  one  series  of  operations  is  sufficient  for 
the  simultaneous  construction  of  u  and  v,  instead  of  one  series  for  u  and 
another  series  for  v  as  in  the  general  case  of  a  complex  function  in  §  6. 
If  this  assumption  be  justified  by  the  same  forms  resulting  from  the  two 
different  methods  of  construction,  it  follows  that  the  two  series  of  opera 
tions,  which  lead  in  the  general  case  to  u  and  to  v,  must  be  equivalent  to 
the  single  series  and  must  therefore  be  connected  by  conditions  ;  that  is,  u 
and  v  as  functions  of  a;  and  y  must  have  their  functional  forms  related. 

We  thus  take 

u  +  iv  —  w  =  f(z)  =  f(x  +  iy) 

without  any  specification  of  the  form  of  f.     When  this  postulated  equation 

is  valid,  we  have 

dw     dw  dz       ,.  ,  .       dw 

_    —  .    _    _      -      I         {  2/9     TTT      _ 

dx  dz  dx  J  ^  '  dz' 
dw  _  dw  "dz  _  .,,.  .  .  dw 
frj  =  ~fad~y~V  (Z)  lfa' 

•  dw     1  dw     dw 

and  therefore  —  =  -—  =  —-  ...........................  (1) 

dx      i  dy      dz 

equations  from  which  the  functional  form  has  disappeared.     Inserting  the 
value  of  w,  we  have 


whence,  after  equating  real  and  imaginary  parts, 

dv  _du        du  _  dv 
dx     dy'      dx     dy" 
These  are  necessary  relations  between  the  functional  forms  of  u  and  v. 

These  relations  are  easily  seen  to  be  sufficient  to  ensure  the  required 
functionality.     For,  on  taking  w  =  ii  +  iv,  the  equations  (2)  at  once  lead  to 

dw  _  1  dw 
dx      i  dy  ' 

,,    ,  .  dw      .dw 

that  is,  to  --  —  \-  1  —  -  =  0, 

ox        dy 


8  RIEMANN'S  [7. 

a  linear  partial  differential  equation  of  the  first  order.  To  obtain  the  most 
general  solution,  we  form  a  subsidiary  system 

dx  _  dy  _  dw 
T==T  ==~0~* 

It  possesses  the  integrals  w,  x  +  iy;  and  then  from  the  known  theory  of 
such  equations  we  infer  that  every  quantity  w  satisfying  the  equation  can  be 
expressed  as  a  function  of  x  +  iy,  i.e.,  of  z.  The  conditions  (2)  are  thus 
proved  to  be  sufficient,  as  well  as  necessary. 

8.  The  preceding  determination  of  the  necessary  and  sufficient  conditions 
of  functional  dependence  is  based  upon  the  existence  of  a  functional  form  ; 
and  yet  that  form  is  not  essential,  for,  as  already  remarked,  it  disappears  from 
the  equations  of  condition.  Now  the  postulation  of  such  a  form  is  equivalent 
to  an  assumption  that  the  function  can  be  numerically  calculated  for  each 
particular  value  of  the  independent  variable,  though  the  immediate  expres 
sion  of  the  assumption  has  disappeared  in  the  present  case.  Experience  of 
functions  of  real  variables  shews  that  it  is  often  more  convenient  to  use 
their  properties  than  to  possess  their  numerical  values.  This  experience  is 
confirmed  by  what  has  preceded.  The  essential  conditions  of  functional 
dependence  are  the  equations  (1),  and  they  express  a  property  of  the  function 

w,  viz.,  that  the  value  of  the  ratio  -r  is  the  same  as  that  of  ~-  ,  or,  in  other 

words,  it  is  independent  of  the  manner  in  which  dz  ultimately  vanishes  by 
the  approach  of  the  point  z  +  dz  to  coincidence  with  the  point  z.  We  are 
thus  led  to  an  entirely  different  definition  of  functionality,  viz.  : 

A  complex  quantity  w  is  a  function  of  another  complex  quantity  z,  when 
they  change  together  in  such  a  manner  that  the  value  of  -,  is  independent  of 
the  value  of  the  differential  element  dz. 

This  is  Riemann's  definition*  ;  we  proceed  to  consider  its  significance. 

We  have 

dw     du  +  idv 
dz      dx  +  idy 

/du      .dv\        dx  /du      .dv\       du 

__     I      __  I        n      _      I      _  _____         I         I     __  L     ^     __      I       _  Y.  _ 

~~  \dx        dxj  dx  +  idy      \dy        dy/  dx  +  idy  ' 
Let  </>  be  the  argument  of  dz  ;  then 


_ 

cos  <£  +  1  sin  </> 


*  Ges.  Werke,  p.  5;  a  modified  definition  is  adopted  by  him,  ib.,  p.  81. 


8.]  DEFINITION    OF   A    FUNCTION 

and  therefore 

dw      .   (du      .dv       .du     dv)  „,.  {du      .dv       .du     dv 

I       I  I     i    n I I      I       1  a—^4>  I  J  I      t  In 

7     —  i  i«i     T"  ^          •  5      •  57  f    •   a**  i<5     Tfc  ^~  ~r  *  « « 

«£      •  [da;        dx        dy     dy}  (dx        dx        dy     dy 

Since  -j—  is  to  be  independent  of  the  value  of  the  differential  element  dz, 
dz 

it  must  be  independent  of  <f>  the  argument  of  dz ;  hence  the  coefficient 
of  e-2*«  in  the  preceding  expression  must  vanish,  which  can  happen  only  if 

du  _dv       dv  _     du 
dx     dy'     dx        dy  " 

These  are  necessary  conditions;  they  are  evidently  also  sufficient  to  make 
^—  independent  of  the  value  of  dz  and  therefore,  by  the  definition,  to  secure 
that  w  is  a  function  of  z. 

By  means  of  the  conditions  (2),  we  have 

dw  _  du      .dv  _dw 
dz       dx        dx      dx  ' 

dw         .du     dv      1  dw 
and  also  — -  =  —  i  - — [_=_. 

dz  dy     dy      i  dy 

agreeing  with  the  former  equations  (1)  and  immediately  derivable  from  the 
present  definition  by  noticing  that  dx  and  idy  are  possible  forms  of  dz. 

It  should  be  remarked  that  equations  (2)  are  the  conditions  necessary 
and  sufficient  to  ensure  that  each  of  the  expressions 

udx  —  vdy  and  vdx  +  udy 

is  a  perfect  differential — a  result  of  great  importance  in  many  investigations 
in  the  region  of  mathematical  physics. 

When  the  conditions  (2)  are  expressed,  as  is  sometimes  convenient,  in 
terms  of  derivatives  with  regard  to  the  modulus  of  z,  say  r,  and  the 
argument  of  z,  say  0,  they  take  the  new  forms 

du_ldv       dv  _     Idu, 

^      —  ~  57j  >       ^~  —  ^TT. (^)- 

or      r  dv       or         r  da 

We  have  so  far  assumed  that  the  function  has  a  differential  coefficient — 
an  assumption  justified  in  the  case  of  functions  which  ordinarily  occur.  But 
functions  do  occur  which  have  different  values  in  different  regions  of  the 

.z-plane,  and  there  is  then  a  difficulty  in  regard  to  the  quantity    ,W  at  the 

boundaries  of  such  regions ;  and  functions  do  occur  which,  though  themselves 
definite  in  value  in  a  given  region,  do  not  possess  a  differential  coefficient  at 
all  points  in  that  region.  The  consideration  of  such  functions  is  not  of 
substantial  importance  at  present :  it  belongs  to  another  part  of  our  subject. 


10  CONFORMAL  [8. 

It  must  not  be  inferred  that,  because  -j-  is  independent  of  the  direction 
in  which  dz  vanishes  when  w  is  a  function  of  z,  therefore  -=-  has  only  one 

value.     The  number  of  its  values  is  dependent  on  the  number  of  values  of  w : 
no  one  of  its  values  is  dependent  on  dz. 

A  quantity,  defined  as  a  function  by  Riemann  on  the  basis  of  this 
property,  is  sometimes*  called  an  analytical  function;  but  it  seems  pre 
ferable  to  reserve  the  term  analytical  in  order  that  it  may  be  associated 
hereafter  (§  34)  with  an  additional  quality  of  the  functions. 

9.  The  geometrical  interpretation  of  complex  variability  leads  to  impor 
tant  results  when  applied  to  two  variables  w  and  z  which  are  functionally 
related. 

Let  P  and  p  be  two  points  in  different  planes,  or  in  different  parts  of 
the  same  plane,  representing  w  and  z  respectively;  and  suppose  that  P  and 
p  are  at  a  finite  distance  from  the  points  (if  any)  which  cause  discontinuity 
in  the  relationship.  Let  q  and  r  be  any  two  other  points,  z  +  dz  and  z  +  8z, 
in  the  immediate  vicinity  of  p ;  and  let  Q  and  E  be  the  corresponding 
points,  w  +  dw  and  w  +  &w,  in  the  immediate  vicinity  of  P.  Then 

dw  j       ^        dw  ? 
dw  =  ^r-  dz.     bw  =  -r—  of, 
dz  dz 

the  value  of  ~  being  the  same  for  both  equations,  because,  as  w  is  a  function 
dz 

of  z,  that  quantity  is  independent  of  the  differential  element  of  z.     Hence 

8w  _  Bz 
dw     dz' 

on  the  ground  that   ,     is  neither  zero  nor  infinite  at  z,  which  is  assumed  not 

CL2 

to  be  a  point  of  discontinuity  in  the  relationship.  Expressing  all  the  differ 
ential  elements  in  terms  of  their  moduli  and  arguments,  let 

dz  =  a-eei,      dw  —  rje^1, 

Sz  =  oV'*,     8w  =  i)<$\ 
and  let  these  values  be  substituted  in  the  foregoing  relation ;  then 

77'      tr 

tj       a 

$-$  =  &-&. 

Hence  the  triangles  QPR  and  qpr  are  similar  to  one  another,  though 
not  necessarily  similarly  situated.  Moreover  the  directions  originally  chosen 
for  pq  and  pr  are  quite  arbitrary.  Thus  it  appears  that  a  functional  relation 

*  Harnack,  §  84. 


1 11 V  -  (<M\*  a.  i^v 

•    I  <a     I    —  I  "5     /      '    \  "> 

$a?/       \oyj       \dy 


9.]  REPRESENTATION   OF   PLANES  11 

between  two  complex  variables  establishes  the  similarity  of  the  corresponding 
infinitesimal  elements  of  those  parts  of  two  planes  which  are  in  the  immediate 
vicinity  of  the  points  representing  the  two  variables. 

The  magnification  of  the  w-plane  relative  to  the  ^-plane  at  the  corre 
sponding  points  P  and  p  is  the  ratio  of  two  corresponding  infinitesimal 

lengths,  say  of  QP  and  qp.     This  is  the  modulus  of  -^— ;  if  it  be  denoted  by 

m,  we  have 

2  _  dw  2 
dz 

_  du  dv      du  dv 
dx  dy     dy  dx ' 

Evidently  the  quantity  m,  in  general,  depends  on  the  variables  and 
therefore  it  changes  from  one  point  to  another ;  hence  the  functional  relation 
between  w  and  z  does  not,  in  general,  establish  similarity  of  finite  parts  of 
the  two  planes  corresponding  to  one  another  through  the  relation. 

It  is  easy  to  prove  that  w  =  az  +  b,  where  a  and  b  are  constants,  is  the 
only  relation  which  establishes  similarity  of  finite  parts ;  and  that,  with  this 
relation,  a  must  be  a  real  constant  in  order  that  the  similar  parts  may  be 
similarly  situated. 

If  u  +  iv  =  w  =  <}>  (z),  the  curves  u  =  constant  and  v  =  constant  cut  at 
right  angles;  a  special  case  of  the  proposition  that,  if  <£  (x  +  iy)  =  u  +  v^, 
where  A,  is  a  real  constant  and  u,  v  are  real,  then  u=  constant  and  v= constant 
cut  at  an  angle  X. 

The  process,  which  establishes  the  infinitesimal  similarity  of  two  planes 
by  means  of  a  functional  relation  between  the  variables  of  the  planes,  may  be 
called  the  conformal  representation  of  one  plane  on  another*. 

The  discussion  of  detailed  questions  connected  with  the  conformal  representation  is 
deferred  until  the  later  part  of  the  treatise,  principally  in  order  to  group  all  such 
investigations  together  ;  but  the  first  of  the  two  chapters,  devoted  to  it,  need  not  be 
deferred  so  late  and  an  immediate  reading  of  some  portion  of  it  will  tend  to  simplify 
many  of  the  explanations  relative  to  functional  relations  as  they  occur  in  the  early 
chapters  of  this  treatise. 

10.  The  analytical  conditions  of  functionality,  under  either  of  the 
adopted  definitions,  are  the  equations  (2).  From  them  it  at  once  follows  that 


8^  +  ty* =     ' 

*  By  Gauss  (Ges.  Werke,  t.  iv,  p.  262)  it  was  styled  conforme  Abbildung,  the  name 
universally  adopted  by  German  mathematicians.  The  French  title  is  representation  conforme ; 
and,  in  England,  Cayley  has  used  orthomorphosis  or  ortliomorphic  transformation. 


12  CONDITIONS   OF   FUNCTIONAL   DEPENDENCE  [10. 

so  that  neither  the  real  nor  the  imaginary  part  of  a  complex  function  can  be 
arbitrarily  assumed. 

If  either  part  be  given,  the  other  can  be  deduced  ;  for  example,  let  u  be 
given  ;  then  we  have 


7  j  j 

dv  =  ^-dx  +  —  dy 

dx          dy 

du  ,       du  j 
=  -=-dx+~-dy, 
dy          ox    ' 

and  therefore,  except  as  to  an  additive  constant,  the  value  of  v  is 

[i    9w  7       du  -,  \ 

-  —  dx  +  5-  dy  I  . 
A    dy         ax   °  I 

In  particular,  when  u  is  an  integral  function,  it  can  be  resolved  into  the 
sum  of  homogeneous  parts 

MI  +  w2  +  w3  +  .  .  .  ; 

and   then,   again  except   as   to   an   additive   constant,   v   can   similarly   be 
expressed  in  the  form 

Vl  +  V2  +  V3  +  ---- 

It  is  easy  to  prove  that 

dum        dum 

™>»  =  y-te-*-ty> 

by  means  of  which  the  value  of  v  can  be  obtained. 

The   case,   when   u   is   homogeneous    of   zero    dimensions,   presents   no 
difficulty  ;  for  we  then  have 


v  =  c-a\ogr,   =c-/f£ 
where  a,  6,  c  are  constants. 

Similarly  for  other  special  cases;  and,  in  the  most  general  case,  only 
a  quadrature  is  necessary. 

The  tests  of  functional  dependence  of  one  complex  on  another  are  of 
effective  importance  in  the  case  when  the  supposed  dependent  complex 
arises  in  the  form  u  +  iv,  where  u  and  v  are  real;  the  tests  are,  of  course, 
superfluous  when  w  is  explicitly  given  as  a  function  of  z.  When  w  does 
arise  in  the  form  u  +  iv  and  satisfies  the  conditions  of  functionality,  perhaps 
the  simplest  method  (other  than  by  inspection)  of  obtaining  the  explicit 
expression  in  terms  of  z  is  to  substitute  z  —  iy  for  x  in  u  +  iv  ;  the  simplified 
result  must  be  a  function  of  z  alone. 

11.  Conversely,  when  w  is  explicitly  given  as  a  function  of  z  and  it  is 
divided  into  its  real  and  its  imaginary  parts,  these  parts  individually 
satisfy  the  foregoing  conditions  attaching  to  u  and  v.  Thus  logr,  where  r 
is  the  distance  of  a  point  z  from  a  point  a,  is  the  real  part  of  log  (z  —  a) 
and  therefore  satisfies  the  equation 


11.] 


EXAMPLE   OF   RIEMANN  S   DEFINITION 


13 


Again,  <f>,  the  angular  coordinate  of  z  relative  to  the  same  point  a,  is 
the  real  part  of  —  i  log  (z  —  a)  and  satisfies  the  same  equation :  the  more 
usual  form  of  <£  being  tan"1  {(y  —  y0)/(®  —  %o)}>  where  a  =  x0  +  iy0.  Again,  if 
a  point  z  be  distant  r  from  a  and  r'  from  b,  then  log  (r/r'\  being  the  real 
part  of  log  {(z  —  a)l(z  —  b)\,  is  a  solution  of  the  same  equation. 

The  following  example,  the  result  of  which  will  be  useful  subsequently*,  uses  the 
property  that  the  value  of  the  derivative  is  independent  of  the  differential  element. 

z-c 


Consider  a  function 


u  +  iv  =  w  =  log 


where  c'  is  the  inverse  of  c  with  regard  to  a  circle  centre  the  origin  0  and  radius  R. 
Then 


z-c 

*       V 

:—  r> 

z-c 


and    the    curves    u  =  constant    are  circles.     Let 

W-     • 
(fig.  4)  Oc  =  r,  xOc  =  a  so  that  c  =  reat,  c'=  —  eal; 

then  if 


Fig.  4. 

the  values  of  X  for  points  in  the  interior  of  the  circle  of  radius  R  vary  from  zero,  when 
circle  u  =  constant  is  the  point  c,  to  unity,  when  the  circle  u  =  constant  is  the  circle  of 
radius  R.  Let  the  point  K  ( =  6eal)  be  the  centre  of  the  circle  determined  by  a  value  of 
X,  and  let  its  radius  be  p  (  =  %MN}.  Then  since 

cM      r  ,.      cN 


we  have 


whence 


r+p-d       r          d  +  P~r 

— Vp-B  Q-p 

r  r 


P  = 


Now  if  dn  be  an  element  of  the  normal  drawn  inwards  at  z  to  the  circle  NzM,  we  have 
dz  =  dx+idy=  —  dn  .  cos  ^  -  idn  .  sin  ^ 

--«*<*», 

where  ^  (  =  zKx'}  is  the  argument  of  z  relative  to  the  centre  of  the  circle.     Hence,  since 

dw        1  1 


we  have 
But 

so  that 

and 


,  .,       ,  du      .dv 

and  therefore        -=-  +  i  -j-  = 
dn        dn 


__  _ 

dz      z  —  c     z-c'1 
du      .dv      dw 
dn 


.dv      dw      /I  1  \   ty 

dn     dn     \z  —  c'     z  —  c) 


e^  -  Reai)  • 


J>  _       1          _  1  !_ 

I  /i!  ~\     ^^      7?  *1^      X  ff  */^ 
\      A7*6      —  J\G  ./t6          i 


*  In  §  217,  in  connection  with  the  investigations  of  Schwarz,  by  whom  the  result  is  stated, 
Ges.  Werke,  t.  ii,  p.  183. 


14  DEFINITIONS  [11. 

Hence,  equating  the  real  parts,  it  follows  that 

du  (_R2-r2A2)2 


dn  ~      \R(R*-  r2)  {E2  -  2Rr\  cos  (^  -  Q)  +  XV2} ' 
the  differential  element  dn  being  drawn  inwards  from  the  circumference  of  the  circle. 

The  application  of  this  method  is  evidently  effective  when  the  curves  u  =  constant, 
arising  from  a  functional  expression  of  w  in  terms  of  z,  are  a  family  of  non-intersecting 
algebraical  curves. 

12.  As  the  tests  which  are  sufficient  and  necessary  to  ensure  that  a 
complex  quantity  is  a  function  of  z  have  been  given,  we  shall  assume  that 
all  complex  quantities  dealt  with  are  functions  of  the  complex  variable 
(§§  6,  7).  Their  characteristic  properties,  their  classification,  and  some  of 
the  simpler  applications  will  be  considered  in  the  succeeding  chapters. 

Some  initial  definitions  and  explanations  will  now  be  given. 

(i).  It  has  been  assumed  that  the  function  considered  has  a  differential 
coefficient,  that  is,  that  the  rate  of  variation  of  the  function  in  any  direction 
is  independent  of  that  direction  by  being  independent  of  the  mode  of  change 
of  the  variable.  We  have  already  decided  (§  8)  not  to  use  the  term  analytical 
for  such  a  function.  It  is  often  called  monogenic,  when  it  is  necessary  to 
assign  a  specific  name ;  but  for  the  most  part  we  shall  omit  the  name,  the 
property  being  tacitly  assumed*. 

We   can  at   once  prove  from  the  definition  that,  when  the  derivative 

/     dw\       •.-.'•,     if-      c      <-•          v  dw      Idw 

w,    = -p-     exists,  it  is  itselt  a  Junction,     .bor  w-,  =-=—  =  -  =—  are  equations 
\     dz )  dx      i  dy 

which,  when  satisfied,  ensure  the  existence  of  w^ ;  hence 

1  dw-!  _  1  3  (dw\ 
i  dy       i  dy  \d%  ) 
_  d_  (I  dw\ 
dx  \i  dyj 
_dw1 
=  l)x  ' 

shewing,  as  in  §  8,  that  the  derivative  ~  is  independent  of  the  direction  in 

CL2 

which  dz  vanishes.     Hence  wl  is  a  function  of  z. 

Similarly  for  all  the  derivatives  in  succession. 

(ii).  Since  the  functional  dependence  of  a  complex  is  ensured  only  if  the 
value  of  the  derivative  of  that  complex  be  independent  of  the  manner  in 
which  the  point  z  +  dz  approaches  to  coincidence  with  z,  a  question  naturally 

*  This  is  in  fact  done  by  Biemann,  who  calls  such  a  dependent  complex  simply  a  function. 
Weierstrass,  however,  has  proved  (§  85)  that  the  idea  of  a  monogenic  function  of  a  complex 
variable  and  the  idea  of  dependence  expressible  by  arithmetical  operations  are  not  coextensive. 
The  definition  is  thus  necessary;  but  the  practice  indicated  in  the  text  will  be  adopted,  as  non- 
monogenic  functions  will  be  of  relatively  rare  occurrence. 


12.]  DEFINITIONS  15 

suggests  itself  as  to  the  effect  on  the  character  of  the  function  that  may  be 
caused  by  the  manner  in  which  the  variable  itself  has  come  to  the  value  of  z. 
If  a  function  have  only  one  value  for  each  given  value  of  the  variable, 
whatever  be  the  manner  in  which  the  variable  has  come  to  that  value,  the 
function  is  called  uniform*.  Hence  two  different  paths  from  a  point  a  to  a 
point  z  give  at  z  the  same  value  for  any  uniform  function ;  and  a  closed 
curve,  beginning  at  any  point  and  completely  described  by  the  ^-variable, 
will  lead  to  the  initial  value  of  w,  the  corresponding  w-curve  being  closed,  if  z 
have  passed  through  no  point  which  makes  w  infinite. 

The  simplest  class  of  uniform  functions  is  constituted  by  algebraical 
rational  functions. 

(iii).  If  a  function  have  more  than  one  value  for  any  given  value  of  the 
variable,  or  if  its  value  can  be  changed  by  modifying  the  path  in  which 
the  variable  reaches  that  given  value,  the  function  is  called  multiform-]'. 
Characteristics  of  curves,  which  are  graphs  of  multiform  functions  corre 
sponding  to  a  2-curve,  will  hereafter  be  discussed. 

One  of  the  simplest  classes  of  multiform  functions  is  constituted  by 
algebraical  irrational  functions. 

(iv).  A  multiform  function  has  a  number  of  different  values  for  the  same 
value  of  z,  and  these  values  vary  with  z :  the  aggregate  of  the  variations  of 
any  one  of  the  values  is  called  a  branch  of  the  function.  Although  the 
function  is  multiform  for  unrestricted  variation  of  the  variable,  it  often 
happens  that  a  branch  is  uniform  when  the  variable  is  restricted  to 
particular  regions  in  the  plane. 

(v).  A  point  in  the  plane,  at  which  two  or  more  branches  of  a  multiform 
function  assume  the  same  value,  is  called  a  branch-point^  of  the  function; 
the  relations  of  the  branches  in  the  immediate  vicinity  of  a  branch-point  will 
hereafter  be  discussed. 

(vi).  A  function  which  is  monogenic,  uniform  and  continuous  over  any 
part  of  the  ^-plane  is  called  holomorphic  §  over  that  part  of  the  plane.  When 
•a  function  is  called  holomorphic  without  any  limitation,  the  usual  implication 
is  that  the  character  is  preserved  over  the  whole  of  the  plane  which  is  not  at 
infinity. 

The  simplest  example  of  a  holomorphic  function  is  a  rational  integral 
algebraical  polynomial. 

*  Also  monodromic,  or  monotropic;  with  German  writers  the  title  is  eindeutig,  occasionally, 
einandrig. 

t  Also  polytropic ;  with  German  writers  the  title  is  mchrdeittig. 

J  Also  critical  point,  which,  however,  is  sometimes  used  to  include  all  special  points  of  a 
function ;  with  German  writers  the  title  is  Verziveigungspunkt,  and  sometimes  Windungspunkt. 
French  writers  use  point  de  ramification,  and  Italians  punto  di  giramento  and  punto  di 
diramazione. 

§  Also  synectic. 


16  EXAMPLES   ILLUSTRATING  [12. 

(vii).  A  root  (or  a  zero)  of  a  function  is  a  value  of  the  variable  for  which 
the  function  vanishes. 

The  simplest  case  of  occurrence  of  roots  is  in  a  rational  integral  alge 
braical  function,  various  theorems  relating  to  which  (e.g.,  the  number  of 
roots  included  within  a  given  contour)  will  be  found  in  treatises  on  the 
theory  of  equations. 

(viii).  The  infinities  of  a  function  are  the  points  at  which  the  value  of 
the  function  is  infinite.  Among  them,  the  simplest  are  the  poles*  of  the 
function,  a  pole  being  an  infinity  such  that  in  its  immediate  vicinity  the 
reciprocal  of  the  function  is  holomorphic. 

Infinities  other  than  poles  (and  also  the  poles)  are  called  the  singular 
points  of  the  function  :  their  classification  must  be  deferred  until  after  the 
discussion  of  properties  of  functions. 

(ix).  A  function  which  is  monogenic,  uniform  and,  except  at  poles, 
continuous,  is  called  a  meromorphic  function  f.  The  simplest  example  is  a 
rational  algebraical  fraction. 

13.  The  following  functions  give  illustrations  of  some  of  the  preceding 
definitions. 

(a)  In  the  case  of  a  meromorphic  function 

F(z) 
111  —  —  *  —  - 

/<*)' 

where  F  and  /  are  rational  algebraical  functions  without  a  common  factor, 
the  roots  are  the  roots  of  F  (z)  and  the  poles  are  the  roots  of  f  (z).  Moreover, 
according  as  the  degree  of  F  is  greater  or  is  less  than  that  of  f,z  =  vo  is  a 
pole  or  a  zero  of  w. 

(b)  If  w  be  a  polynomial  of  order  n,  then  each  simple  root  of  w  is  a 

branch-point  and  a  zero  of  wm,  where  m  is  a  positive  integer  ;  z  =  oo  is 
a  pole  of  w;  and  z=  oo  is  a  pole  but  not  a  branch-point  or  is  an  infinity 
(though  not  a  pole)  and  a  branch-point  of  w$  according  as  n  is  even  or  odd. 

(c)  In  the  case  of  the  function 

1 


w- 


sn- 

z 


(the  notation  being  that  of  Jacobian  elliptic  functions),  the  zeros  are  given  by 


z 
for  all  positive  and  negative  integral  values  of  m  and  of  m'.     If  we  take 


-  =  iK'  +  2mK  +  Zm'iK'  -f  £ 

z 

*  Also  polar  discontinuities  ;  also  (§  32)  accidental  singularities. 

t  Sometimes  rey-nlar,  but  this  term  will  be  reserved  for  the  description  of  another  property  of 
functions. 


13.]  THE   DEFINITIONS  17 

where  £  may  be  restricted  to  values  that  are  not  large,  then 

w  =  (-  l)m  &sn£ 

so  that,  in  the  neighbourhood  of  a  zero,  w  behaves  like  a  holomorphic 
function.  There  is  evidently  a  doubly-infinite  system  of  zeros:  they  are 
distinct  from  one  another  except  at  the  origin,  where  an  infinite  number 
practically  coincide. 

The  infinities  of  w  are  given  by 


for  all  positive  and  negative  integral  values  of  n  and  of  n'.     If  we  take 


-  =  2nK  +  Zn'iK'  +  £ 

2! 

then  -  =  (-l)"sn£ 

w 

so  that,  in  the  immediate  vicinity  of  f=0,  -  is  a  holomorphic   function. 

Hence  f  =  0  is  a  pole  of  w.  There  is  thus  evidently  a  doubly-infinite  system 
of  poles  ;  they  are  distinct  from  one  another  except  at  the  origin,  where  an 
infinite  number  practically  coincide.  But  the  origin  is  not  a  pole;  the 
function,  in  fact,  is  there  not  determinate,  for  it  has  an  infinite  number  of 
zeros  and  an  infinite  number  of  infinities,  and  the  variations  of  value  are  not 
necessarily  exhausted. 

For  the  function  —  j  ,  the  origin  is  a  point  which  will  hereafter  be  called 

sn- 

z 

an  essential  singularity. 


F. 


CHAPTER  II. 

INTEGRATION  OF  UNIFORM  FUNCTIONS. 

14.  THE  definition  of  an  integral,  that  is  adopted  when  the  variables 
are  complex,  is  the  natural  generalisation  of  that  definition  for  real  variables 
in  which  it  is  regarded  as  the  limit  of  the  sum  of  an  infinite  number  of 
infinitesimally  small  terms.  It  is  as  follows : — 

Let  a  and  z  be  any  two  points  in  the  plane ;  and  let  them  be  connected 
by  a  curve  of  specified  form,  which  is  to  be  the  path  of  variation  of  the 
independent  variable.  Let  f(z)  denote  any  function  of  0;  if  any  infinity 
of  f(z)  lie  in  the  vicinity  of  the  curve,  the  line  of  the  curve  will  be  chosen 
so  as  not  to  pass  through  that  infinity.  On  the  curve,  let  any  number  of 
points  z^  z2,...,  zn  in  succession  be  taken  between  a  and  z ;  then,  if  the  sum 

(z, -  a)f  (a)  +  (z, -  z,}  f  (z,)  +  ...  +  (z- zn)f(zn} 

have  a  limit,  when  n  is  indefinitely  increased  so  that  the  infinitely  numerous 
points  are  in  indefinitely  close  succession  along  the  whole  of  the  curve  from  a 
to  z,  that  limit  is  called  the  integral  of  /  (z)  between  a  and  z.  It  is  denoted, 
as  in  the  case  of  real  variables,  by 


f(z)dz. 

The  limit,  as  the  value  of  the  integral,  is  associated  with  a  particular 
curve :  in  order  that  the  integral  may  have  a  definite  value,  the  curve  (called 
the  path  of  integration)  must,  in  the  first  instance,  be  specified*.  The 
integral  of  any  function  whatever  may  not  be  assumed  to  depend  in  general 
only  upon  the  limits. 

15.     Some  inferences  can  be  made  from  the  definition. 
(I.)     The  integral  along  any  path  from  a  to  z  passing  through  a  point  £  is 
the  sum  of  the  integrals  from  a  to  £  and  from  \  to  z  along  the  same  path. 

*  This  specification  is  tacitly  supplied  when  the  variables  are  real :  the  variable  point  moves 
along  the  axis  of  x. 


15.]  INTEGRATION  19 

Analytically,  this  is  expressed  by  the  equation 

P  /  (*)  dz  =  I  V  (*)  dz  +  I  V  (*)  <fc, 

^  a  J  a  J  f 

the  paths  on  the  right-hand  side  combining  to  form  the  path  on  the  left. 

(II.)      When  the  path  is  described  in  the  reverse  direction,  the  sign  of  the 
integral  is  changed  :  that  is, 


the  curve  of  variation  between  a  and  z  being  the  same. 

(III.)  The  integral  of  the  sum  of  a  finite  number  of  terms  is  equal  to 
the  sum  of  the  integrals  of  the  separate  terms,  the  path  of  integration  being 
the  same  for  all. 

(IV.)  If  a  function  f  (z)  be  finite  and  continuous  along  any  finite  line 
between  two  points  a  and  z,  the  integral  \  f(z)dz  is  finite. 

J  a 

Let  7  denote  the  integral,  so  that  we  have  I  as  the  limit  of 


r=0 

hence  |/|  =  limit  of 


Because  f(z}  is  finite  and  continuous,  its  modulus  is  finite  and  therefore 
must  have  a  superior  limit,  say  M,  for  points  on  the  line.     Thus 


80  that  I/I  <  limit  of  r+1 

<MS, 

where  8  is  the  finite  length  of  the  path  of  integration.     Hence  the  modulus 
of  the  integral  is  finite  ;  the  integral  itself  is  therefore  finite. 

No  limitation  has  been  assigned  to  the  path,  except  finiteness  in  length  ; 
the  proposition  is  still  true  when  the  curve  is  a  closed  curve  of  finite  length. 

Hermite  and  Darboux  have  given  an  expression  for  the  integral  which 
leads  to  the  same  result.     We  have  as  above 


f(z)\  dz\, 
where  6  is  a  real  positive  quantity  less  than  unity.    The  last  integral  involves 


2—2 


20  THEOREMS  [15. 

only  real  variables;  hence*  for  some  point  £  lying  between  a  and  z,  we  have 


f 

J  a 


so  that  l/|  =  fl9f|/(!)|. 

It  therefore  follows  that  there  is  some  argument  a  such  that,  if  X  =  Be10-, 


This  form  proves  the  finiteness  of  the  integral  ;  and  the  result  is  the 
generalisation  f  to  complex  variables  of  the  theorem  just  quoted  for  real 
variables. 

(V.)  When  a,  function  is  expressed  in  the  form  of  a  series,  which  converges 
uniformly  and  unconditionally,  the  integral  of  the  function  along  any  path  of 
finite  length  is  the  sum  of  the  integrals  of  the  terms  of  the  series  along  the 
same  path,  provided  that  path  lies  within  the  circle  of  convergence  of  the  series  : 
—  a  result,  which  is  an  extension  of  (III.)  above. 

Let  M0  +  MI  +  u.2  +  .  .  .  be  the  converging  series  ;  take 

/  (z)  =  U0  +  M!  +  .  .  .  +  Un  +  R, 

where  \R\  can  be  made  infinitesimally  small  with  indefinite  increase  of  n, 
because  the  series  converges  uniformly  and  unconditionally.  Then  by  (III.), 
or  immediately  from  the  definition  of  the  integral,  we  have 

rz  rs  rz  rz  re 

f(z)dz=  I    u0dz  +      ^dz  +  .  ..  +  I   undz  +  1    Rdz, 

J  a  J  a  J  a  J  a  J  a 

the  path  of  integration  being  the  same  for  all  the  integrals.     Hence,  if 

re  n     re 

(S)  =  I    f  (z)  dz  —  2   I   umdz, 

J  a  m=oJ  a 

ft 

we  have  ©  =  I    Rdz. 


ft 

=  I 

J  a 


Let  R  be  the  greatest  value  of  \R\  for  points  in  the  path  of  integration 
from  a  to  z,  and  let  8  be  the  length  of  this  path,  so  that  8  is  finite  ; 

then,  by  (IV.), 

\®\<SR. 

Now  8  is  finite  ;  and,  as  n  is  increased  indefinitely,  the  quantity  R  tends 
towards  zero  as  a  limit  for  all  points  within  the  circle  of  convergence  and 
therefore  for  all  points  on  the  path  of  integration  provided  that  the  path  lie 
within  the  circle  of  convergence.  When  this  proviso  is  satisfied,  |@|  becomes 
infinitesimally  small  and  therefore  also  ®  becomes  infinitesimally  small  with 

*  Todhunter's  Integral  Calculus  (4th  ed.),  §  40;  Williamson's  Integral  Calculus,  (Gth  ed.),  §  96. 
t  Hermite,  Cours  d  la  faculte  dcs  sciences  de  Paris  (46mc  ed.,  1891),  p.  59,  where  the  reference 
to  Darboux  is  given. 


15.] 


ON   INTEGRATION 


21 


indefinite  increase  of  n.     Hence,  under  the  conditions  stated  in  the  enuncia 
tion,  we  have 

rs  oo     r% 

f(z)dz-  2   I  umdz  =  0, 

J  a  m^QJ  a 

which  proves  the  proposition. 

16.     The  following  lemma*  is  of  fundamental  importance. 

Let  any  region  of  the  plane,  on  which  the  ^-variable  is  represented,  be 
bounded  by  one  or  more  simple^  curves  which  do  not  meet  one  another: 
each  curve  that  lies  entirely  in  the  finite  part  of  the  plane  will  be  considered 
to  be  a  closed  curve. 

If '  p  and  q  be  any  two  functions  of  cc  and  y,  which,  for  all  points  within  the 
region  or  along  its  boundary,  are  uniform,  finite  and  continuous,  then  the 
integral 

fffdq     dp\j    , 
1 1    a    -  a     dxdy, 

JJ  \dx     dyj 

extended  over  the  whole  area  of  the  region,  is  equal  to  the  integral 

f(pdx  +  qdy), 
taken  in  a  positive  direction  round  the  whole  boundary  of  the  region. 

(As  the  proof  of  the  proposition  does  not  depend  on  any  special  form  of 
region,  we  shall  take  the  area  to  be  (fig.  5)  that  which  is  included  by  the 
curve  QiPiQs'Pa'  and  excluded  by  P^Qz'PsQs  and  excluded  by  P/P2.  The 
positive  directions  of  description  of  the  curves  are  indicated  by  the  arrows ; 
and  for  integration  in  the  area  the  positive  directions  are  those  of  increas 
ing  a;  and  increasing  y.) 


AB 


Fig.  5. 

*  It  is  proved  by  Eiemann,  Ges.  Werke,  p.  12,  and  is  made  by  him  (as  also  by  Cauchy)  the 
basis  of  certain  theorems  relating  to  functions  of  complex  variables. 

t  A  curve  is  called  simple,  if  it  have  no  multiple  points.  The  aim,  in  constituting  the  boundary 
from  such  curves  is  to  prevent  the  superfluous  complexity  that  arises  from  duplication  of  area  on 
the  plane.  If,  in  any  particular  case,  multiple  points  existed,  the  method  of  meeting  the  difficulty 
would  be  to  take  each  simple  loop  as  a  boundary. 


22  FUNDAMENTAL   THEOREM  [16. 

First,  suppose  that  both  p  and  q  are  real.     Then,  integrating  with  regard 
to  x,  we  have  * 


where  the  brackets  imply  that  the  limits  are  to  be  introduced.  When  the 
limits  are  introduced  along  a  parallel  GQ^...  to  the  axis  of  x,  then,  since 
CQiQi'.  •  •  gives  the  direction  of  integration,  we  have 

[qdy]  =  -  qjdyj.  +  qi'dt/i  -  q.2dy2  +  q-2'dy2'  -  q3dy3  +  q»dy9', 

where  the  various  differential  elements  are  the  projections  on  the  axis  of  y 
of  the  various  elements  of  the  boundary  at  points  along  GQiQJ.... 

Now  when  integration  is  taken  in  the  positive  direction  round  the  whole 
boundary,  the  part  of  /  qdy  arising  from  the  elements  of  the  boundary  at  the 
points  on  CQjQ/...  is  the  foregoing  sum.  For  at  Q3'  it  is  qa'dy3  because  the 
positive  element  dy9,  which  is  equal  to  CD,  is  in  the  positive  direction  of 
boundary  integration;  at  Q3  it  is  —q3dys  because  the  positive  element  dy3, 
also  equal  to  CD.  is  in  the  negative  direction  of  boundary  integration  ; 
at  Qz  it  is  q2'dy2',  for  similar  reasons  ;  at  Q.2  it  is  —  q2dya,  for  similar  reasons  ; 
and  so  on.  Hence 


corresponding  to  parallels  through  C  and  D  to  the  axis  of  x,  is  equal  to 
the  part  of  fqdy  taken  along  the  boundary  in  the  positive  direction  for  all 
the  elements  of  the  boundary  that  lie  between  those  parallels.  Then  when 
we  integrate  for  all  the  elements  CD  by  forming  f[qdy],  an  equivalent  is 
given  by  the  aggregate  of  all  the  parts  of  fqdy  taken  in  the  positive  direction 
round  the  whole  boundary  ;  and  therefore 


on  the  suppositions  stated  in  the  enunciation. 
Again,  integrating  with  regard  to  y,  we  have 


when  the  limits  are  introduced  along  a  parallel  RP^P^. . .  to  the  axis  of  y : 
the  various  differential  elements  are  the  projections  on  the  axis  of  x  of  the 
various  elements  of  the  boundary  at  points  along  SPjP/.... 

It  is  proved,  in  the  same  way  as  before,  that  the  part  of  -  jpdx  arising 
from  the  positively-described  elements  of  the  boundary  at  the  points  on 
BP^'...  is  the  foregoing  sum.  At  P3  the  part  of  fpdac  is  -  p3'dx3,  because 
the  positive  element  dx3,  which  is  equal  to  AB,  is  in  the  negative  direction 

*  It  is  in  this  integration,  and  in  the  corresponding  integration  for  p,  that  the  properties  of 
the  function  q  are  assumed :  any  deviation  from  uniformity,  finiteness  or  continuity  within  the 
region  of  integration  would  render  necessary  some  equation  different  from  the  one  given  in 
the  text. 


16.]  IN    INTEGRATION  23 

of  boundary  integration  ;  at  P3  it  is  p3dx3,  because  the  positive  element 
dx3,  also  equal  to  AB,  is  in  the  positive  direction  of  boundary  integration; 
and  so  on  for  the  other  terms.  Hence 

-  [pdas], 

corresponding  to  parallels  through  A  and  B  to  the  axis  of  y,  is  equal  to 
the  part  of  fpdx  taken  along  the  boundary  in  the  positive  direction  for  all 
the  elements  of  the  boundary  that  lie  between  those  parallels.  Hence 
integrating  for  all  the  elements  AB,  we  have  as  before 

[[dp  j  j  ,   j 

~  dxdy  =  —  I  pax, 

JJdy 

and  therefore  II  U        ?r  )  dxdy=f(pdx  +  qdy). 

Secondly,  suppose  that  p  and  q  are  complex.  When  they  are  resolved 
into  real  and  imaginary  parts,  in  the  forms  p'  +  ip"  and  q'  +  iq"  respectively, 
then  the  conditions  as  to  uniformity,  finiteness  and  continuity,  which  apply  to 
p  and  q,  apply  also  to  p',  q',  p",  q".  Hence 


and  ~  -     -   dxdy  =  j(p"dx  +  q"dy), 

and  therefore  1  1  [  2*  _  J9  j  dxdy  =  J(pdx  +  qdy} 

JJ  \ox     oy/ 

which  proves  the  proposition. 

No  restriction  on  the  properties  of  the  functions  p  and  q  at  points 
that  lie  without  the  region  is  imposed  by  the  proposition.  They  may  have 
infinities  outside,  they  may  cease  to  be  continuous  at  outside  points  or  they 
may  have  branch-points  outside  ;  but  so  long  as  they  are  finite  and  continuous 
everywhere  inside,  and  in  passing  from  one  point  to  another  always  acquire 
at  that  other  the  same  value  whatever  be  the  path  of  passage  in  the  region, 
that  is,  so  long  as  they  are  uniform  in  the  region,  the  lemma  is  valid. 

17.     The  following  theorem  due  to  Cauchy*  can  now  be  proved  :  _ 
If  a  function  f(z)  be  holomorphic  throughout  any  region  of  the  z-plane, 
then  the  integral  ff(z)  dz,  taken  round  the  whole  boundary  of  that  region,  is  zero. 
We  apply  the  preceding  result  by  assuming 

p=f(z\   q  =  ip  =  if(z); 

owing  to  the  character  of  f(z),  these  suppositions  are  consistent   with  the 

*  For  an  account  of  the  gradual  development  of  the  theory  and,  in  particular,  for  a 
statement  of  Cauchy's  contributions  to  the  theory  (with  references),  see  Casorati,  Teorica 
delle  funzioni  di  variabili  complcsse,  pp.  64-90,  102-106.  The  general  theory  of  functions, 
as  developed  by  Briot  and  Bouquet  in  their  treatise  Theoric  des  fonctiom  ellipUques,  is  based 
upon  Cauchy's  method. 


24  INTEGRATION   OF  [17. 

conditions  under  which  the  lemma  is  valid.     Since  p  is  a  function  of  z,  we 
have,  at  every  point  of  the  region, 

dp  _  I  dp 

das      i  dy  ' 
and  therefore,  in  the  present  case, 

dq  _  .  dp  _  dp 

das        doc     dy  ' 

There  is  no  discontinuity  or  infinity  of  p  or  q  within  the  region  ;  hence 


the  integral  being  extended  over  the  region.     Hence  also 

!(pdx  +  qdy)  =  0,      A^    ^/ 
when  the  integral  is  taken  round  the  whole  boundary  of  the  region.     But 

pdx  +  qdy  =  pdx  +  ipdy 
—  pdz 
=f(z)dz, 

and  therefore  //(X)  dz  =  0, 

the  integral  being   taken  round  the  whole  boundary  of  the  region   within 
which  f(z)  is  holomorphic. 

It  should  be  noted  that  the  theorem  requires  no  limitation  on  the  cha 
racter  of/(^)  for  points  z  that  are  not  included  in  the  region. 

Some  important  propositions  can  be  derived  by  means  of  the  theorem,  as 
follows. 

18.     When  a  function  f  (z)  is  holomorphic  over  any  continuous  region 

rz 
of  the  plane,  the  integral  I  f(z)dz  is  a  holomorphic  function  of  2  provided  the 

J  a 

points  z  and  a  as  well  as  the  whole  path  of  integration  lie  within  that  region. 

The  general  definition  (§  14)  of  an  integral  is  associated  with  a  specified 
path  of  integration.  In  order  to  prove  that  the  integral  is  a  holomorphic 
function  of  z,  it  will  be  necessary  to  prove  (i)  that  the  integral  acquires  the 
same  value  in  whatever  way  the  point  z  is  attained,  that  is,  that  the  value  is 
independent  of  the  path  of  integration,  (ii)  that  it  is  finite,  (iii)  that  it 
is  continuous,  and  (iv)  that  it  is  monogenic. 

Let  two  paths  ayz  and  afiz  between  a  and  z  be  drawn  (fig.  6)  in  the 
continuous    region    of   the    plane    within   which  f(z)  is 
holomorphic.     The  line  ayzfia  is  a  contour  over  the  area 
of  which  /  (z}  is    holomorphic  ;   and  therefore  ff(z)  dz 
vanishes    when    the    integral    is    taken    along    ayzfta. 
Dividing  the  integral  into  two  parts  and  implying  by 
Zy,  Zp  that  the  point  z  has  been  reached  by  the  paths     a" 
a<yz,  a{3z  respectively,  we  have  Fig.  6. 


18.]  HOLOMOEPHIC   FUNCTIONS  25 


and  therefore  */  (z)  dz  =  -      f  (z)  dz 

J  a  J  Zg 

-?/*/(*)* 

J  a 
Thus  the  value  of  the  integral  is  independent  of  the  way  in  which  z  has 

FZ 

acquired  its  value  ;  and  therefore  I   f(z)  dz  is  uniform  in  the  region.     Denote 
it  by  F(z). 

Secondly,  f(z)  is  finite  for  all  points  in  the  region  and,  after  the  result 
of  §  17,  we  naturally  consider  only  such  paths  between  a  and  z  as  are  finite  in 
length,  the  distance  between  a  and  z  being  finite;  hence  (§  15,  IV.)  the 
integral  F  (z}  is  finite  for  all  points  z  in  the  region. 

Thirdly,  let  z'  (=  z  4-  82)  be  a  point  infinitesimally  near  to  z  ;  and  consider 
I  f(z)  dz.  By  what  has  just  been  proved,  the  path  from  a  to  z'  can  be  taken 

J  d 

aftzz'  ;  therefore 

(*/(*)  dz  =  [/(z)  dz  +  lZf(z)  dz 

J  *  J  a  J  z 

fz+8z  rz  rz+Sz 

or  f(z}dz-  \  f(z)dz=\         f(z)dz, 

J  a  J  a  J  z 

fz+Sz 

80  that  F(z  +  Sz)  -  F(z)  =  f(z}  dz. 

J  2 

Now  at  points  in  the  infinitesimal  line  from  z  to  z'  ,  the  value  of  the 
continuous  function  f(z)  differs  only  by  an  infinitesimal  quantity  from  its 
value  at  z  ;  hence  the  right-hand  side  is 


where  e|  is  an  infinitesimal  quantity  vanishing  with  ck     It  therefore  follows 
that 


is    an    infinitesimal    quantity  with    a    modulus  of  the  same  order  of  small 
quantities  as  \Sz\.     Hence  F  (z)  is  continuous  for  points  z  in  the  region. 
Lastly,  we  have 


and  therefore  F(z  +  Sz)-F(z) 

82 

has  a  limit  when  Sz  vanishes;  and  this  limit,  f(z),  is  independent  of  the 
way  in  which  8z  vanishes.  Hence  F  (z)  has  a  differential  coefficient ;  the 
integral  is  monogenic  for  points  z  in  the  region. 


26  INTEGRATION    OF  [18. 

Hence  F  (z),  which  is  equal  to 

*  f(z)d*t 


is  uniform,  finite,  continuous  and  monogenic;  it  is  therefore  a  holomorphic 
function  of  z. 

As  in  §  16  for  the  functions  p  and  q,  so  here  for  f(z),  no  restriction  is 
placed  on  properties  of / (z)  at  points  that  do  not  lie  within  the  region;  so 
that  elsewhere  it  may  have  infinities,  or  discontinuities  or  branch  points. 
The  properties,  essential  to  secure  the  validity  of  the  proposition,  are 
(i)  that  no  infinities  or  discontinuities  lie  within  the  region,  and  (ii)  that  the 
same  value  of  f(z)  is  acquired  by  whatever  path  in  the  continuous  region 
the  variable  reaches  its  position  z. 

COROLLARY.  No  change  is  caused  in  the  value  of  the  integral  of  a 
holomorphic  function  between  two  points  when  the  path  of  integration  between 
the  points  is  deformed  in  any  manner,  provided  only  that,  during  the  defor 
mation,  no  part  of  the  path  passes  outside  the  boundary  of  the  region  within 
which  the  function  is  holomorphic. 

This  result  is  of  importance,  because  it  permits  special  forms  of  the  path 
of  integration  without  affecting  the  value  of  the  integral. 

19.  When  a  function  f(z)  is  holomorphic  over  a  part  of  the  plane 
bounded  by  two  simple  curves  (one  lying  within  the  other),  equal  values  of 
ff(z)  dz  are  obtained  by  integrating  round  each  of  the  curves  in  a  direction, 
which — relative  to  the  area  enclosed  by  each — is  positive. 

The  ring-formed  portion  of  the  plane  (fig.  1,  p.  3)  which  lies  between 
the  two  curves  being  a  region  over  which  f(z)  is  holomorphic,  the  integral 
ff(z)  dz  taken  in  the  positive  sense  round  the  whole  of  the  boundary  of 
the  included  portion  is  zero.  The  integral  consists  of  two  parts :  first,  that 
round  the  outer  boundary  the  positive  sense  of  which  is  DEF',  and  second, 
that  round  the  inner  boundary  the  positive  sense  of  which  for  the  portion  of 
area  between  ABC  and  DEF  is  ACE.  Denoting  the  value  of  ff(z)dz  round 
DEF  by  (DEF),  and  similarly  for  the  other,  we  have 

(ACB)  +  (DEF)  =  0. 

The  direction  of  an  integral  can  be  reversed  if  its  sign  be  changed,  so  that 
(ACB)  =  -  (ABC)  ;  and  therefore 

(ABC)  =  (DEF). 

But  (ABC)  is  the  integral  ff(z)dz  taken  round  ABC,  that  is,  round  the 
curve  in  a  direction  which,  relative  to  the  area  enclosed  by  it,  is  positive. 

The  proposition  is  therefore  proved. 

The  remarks  made  in  the  preceding  case  as  to  the  freedom  from  limitations 
on  the  character  of  the  function  outside  the  portion  are  valid  also  in  this  case. 


19.]  HOLOMORPHIC    FUNCTIONS  27 

COROLLARY  I.  When  the  integral  of  a  function  is  taken  round  the  whole 
of  any  simple  curve  in  the  plane,  no  change  is  caused  in  its  value  by  continuously 
deforming  the  curve  into  any  other  simple  curve  provided  that  the  function 
is  holomorphic  over  the  part  of  the  plane  in  which  the  deformation  is  effected. 

COROLLARY  II.  When  a  function  f  (z)  is  holomorphic  over  a  continuous 
portion  of  a  plane  bounded  by  any  number  of  simple  non-intersecting  curves, 
all  but  one  of  which  are  external  to  one  another  and  the  remaining  one  of 
which  encloses  them  all,  the  value  of  the  integral  jf(z)  dz  taken  positively  round 
the  single  external  curve  is  equal  to  the  sum  of  the  values  taken  round  each  of 
the  other  curves  in  a  direction  which  is  positive  relative  to  the  area  enclosed 
by  it. 

These  corollaries  are  of  importance  in  finding  the  value  of  the  integral 
of  a  meromorphic  function  round  a  curve  which  encloses  one  or  more  of  the 
poles.  The  fundamental  theorem  for  such  integrals,  also  due  to  Cauchy,  is 
the  following. 

20.  Let  f(z)  denote  a  function  which  is  holomorphic  over  any  region  in 
the  z-plane  and  let  a  denote  any  point  within  that  region,  which  is  not  a  zero 

°ff(2);  then 

.,  ,       1     f/0)    , 

f(a)  =  ^—  •     *-*-*  az> 

2vnJ  z-a 


the  integral  being  taken  positively  round  the  whole  boundary  of  the  region. 

With  a  as  centre  and  a  very  small  radius  p,  describe  a  circle  G,  which  will 
be  assumed  to  lie  wholly  within  the  region;  this  assumption  is  justifiable 
because  the  point  a  lies  within  the  region.  Because  f  (z)  is  holomorphic  over 
the  assigned  region,  the  f  unction  f(z)l(z  —  a)  is  holomorphic  over  the  whole  of 
the  region  excluded  by  the  small  circle  C.  Hence,  by  Corollary  II.  of  §  19,  we 
have 


z-a 


the  notation  implying  that    the    integrations   are    taken    round    the    whole 
boundary  B  and  round  the  circumference  of  G  respectively. 

For  points  on  the  circle  C,  let  z  —  a  =  peei,  so  that  9  is  the  variable  for 
the  circumference  and  its  range  is  from  0  to  2?r  ;  then  we  have 

dz 


z  —  a 


=  id6. 


Along  the  circle  f(z)=f  (a  +  peei) ;  the  quantity  p  is  very  small  and  /  is 
finite  and  continuous  over  the  whole  of  the  region  so  that  f(a  +  peei)  differs 
from  /(«)  only  by  a  quantity  which  vanishes  with  p.  Let  this  difference 
be  e,  which  is  a  continuous  small  quantity;  then  |ej  is  a  small  quantity 
which,  for  every  point  on  the  circumference  of  C,  vanishes  with  p.  Then 


28  INTEGRATION   OF  [20. 


"  edO. 
o 


If  E  denote  the  value  of  the  integral  on  the  right-hand  side,  and  77  the 
greatest  value  of  the  modulus  of  e  along  the  circle,  then,  as  in  §  15, 

/•2ir 

i  E  <        I  e  d6 


f 


Now  let  the  radius  of  the  circle  diminish  to  zero:  then  77  also  diminishes 
to  zero  and  therefore  E  ,  necessarily  positive,  becomes  less  than  any  finite 
quantity  however  small,  that  is,  E  is  itself  zero;  and  thus  we  have 


z  —  a 
which  proves  the  theorem. 

This   result    is    the    simplest    case    of    the    integral    of    a    meromorphic 

f(z} 
function.     The  subject  of  integration  is  —  —  ,  a  function  which  is  monogenic 

and  uniform  throughout  the  region  and  which,  everywhere  except  at  z  =  a,  is 
finite  and  continuous  ;  moreover,  z  =  a  is  a  pole,  because  in  the  immediate 

Z  ~~~  CL 

vicinity  of  a  the  reciprocal  of  the  subject  of  integration,  viz.  ^-rr  >  i-B  h°l°- 
morphic. 

The  theorem  may  therefore  be  expressed  as  follows  : 

If  g  (z)  be  a  meromorphic  function,  which  in  the  vicinity  of  a  can  be 

f(z} 
expressed  in  the  form  J         where  f(a)  is  not  zero  and  which  at  all  other 

Z  —  CL 

points  in  a  region  enclosing  a  is  holomorphic,  then 

-  —  .  fg  (z)  dz  =  limit  of  (z  —  a)g  (z)  when  z  —  a, 


the  integral  being  taken  round  a  curve  in  the  region  enclosing  the  point  a. 

The  pole  a  of  the  function  g  (z)  is  said  to  be  simple,  or  of  the  first  order, 
or  of  multiplicity  unity. 

Corollary.  The  more  general  case  of  a  meromorphic  function  with  a 
finite  number  of  poles  can  easily  be  deduced.  Let  these  be  a1}...,  an  each 
assumed  to  be  simple  ;  and  let 

G  (z)  =  (z-  a,)  (z  -  aa).  ..(z  -  an). 


20.]  MEROMORPHIC   FUNCTIONS  29 

Let  f(z)  be  a  holomorphic  function  within  a  region  of  the  2-plane  bounded 
by  a  simple  contour  enclosing  the  n  points  a1}  a»,...an,  no  one  of  which  is  a 
zero  off(z).  Then  since 


f(z)        »        1       f(z) 

we  have  j^~(  =  S   „,,    .  -^--  . 

6r  (#)      r=i  Or  (ar)  z  —  ar 

w    ^      f       u  f/(*),j        3       !       f/(*)  ,7 

We  therefore  have  "L  ,  '  dz  =  2<  >..  ,    .  I  dz, 

J  &(*)          r=iCr  (ar)J  2-ar 

each  integral  being  taken  round  the  boundary.     But  the  preceding  proposition 
gives 


because  f(z)  is  holomorphic  over  the  whole  region  included  in  the  contour  ; 
and  therefore 


the  integral  on  the  left-hand  side  being  taken  in  the  positive  direction*. 

The  result  just  obtained  expresses  the  integral  of  the  meromorphic 
function  round  a  contour  which  includes  a  finite  number  of  its  simple  poles. 
It  can  be  otherwise  obtained  by  means  of  Corollary  II.  of  §  19,  by  adopting 
a  process  similar  to  that  adopted  above,  viz.,  by  making  each  of  the  curves  in 
the  Corollary  quoted  small  circles  round  the  points  Oj,...,  an  with  ultimately 
vanishing  radii. 

21.  The  preceding  theorems  have  sufficed  to  evaluate  the  integral  of 
a  function  with  a  number  of  simple  poles  :  we  now  proceed  to  obtain 
further  theorems,  which  can  be  used  among  other  purposes  to  evaluate 
the  integral  of  a  function  with  poles  of  order  higher  than  the  first. 

We  still  consider  a  function  f(z)  which  is  holomorphic  within  a  given 
region.  Then,  if  a  be  a  point  within  the  region  which  is  not  a  zero  of  f(z), 
we  have 


z  -  a 


the    point  a   being   neither  on  the    boundary   nor   within    an   infinitesimal 
distance  of  it.     Let  a  +  Sa  be  any  other  point  within  the  region  ;  then 

dz, 


z  —  a  —  8a 

*  We  shall  for  the  future  assume  that,  if  no  direction  for  a  complete  integral  be  specified,  the 
positive  direction  is  taken. 


30 

and  therefore 


PROPERTIES   OF 


[21. 


iff, 


8a 


f(z)dz, 


t  J  ((*  -  a)2     (z  -  of  (z-a  -Sa)j 
the  integral  being  in  every  case  taken  round  the  boundary. 

Since  f(z)  is  monogenic,  the  definition  of  /'(a),  the  first  derivative  of 
/(a),  gives  /'(a)  as  the  limit  of 

f(a  +  Ba)-f(a) 
Ba 

when  Ba  ultimately  vanishes  ;  hence  we  may  take 


where  a  is  a  quantity  which  vanishes  with  Ba  and  is  therefore  such  that  \  a  \ 
also  vanishes  with  Ba.     Hence 


dividing  out  by  Sa  and  transposing,  we  have 


As  yet,  there  is  no  limitation  on  the  value  of  Sa  ;  we  now  proceed  to  a  limit 
by  making  a  +  Ba  approach  to  coincidence  with  a,  viz.,  by  making  Ba 
ultimately  vanish.  Taking  moduli  of  each  of  the  members  of  the  last 
equation,  we  have 


(a)  _  i  f  J(* 

2in  j  (z  -  o 


_„  +  ««. 


(z  —  a)2  (z  —  a  —  Ba) 


27T 


dz 


Let  the  greatest  modulus  of  -. ~ =r—.  for  points  z  along  the 

(j  —  a)2  (z  —  a  —  Ba) 

boundary  be  M,  which  is  a  finite  quantity  on  account  of  the  conditions 
applying  to  f(z)  and  the  fact  that  the  points  a  and  a  +  Ba  are  not 
infinitesimally  near  the  boundary.  Then,  by  §  15, 


t 


dz 


'0-a)2  (z-a-Ba) 

<MS, 

where  8  is  the  whole  length  of  the  boundary,  a  finite  quantity.     Hence 

1    f  f(z}     ,        ,         |8a| 


dz 


c 

ITT 


21.]  HOLOMORPHIC    FUNCTIONS  31 

When  we  proceed  to  the  limit  in  which  Sa  vanishes,  we  have  Ba  =  0 
and  |o-|  =  0,  ultimately;  hence  the  modulus  on  the  left-hand  side  ultimately 
vanishes  and  therefore  the  quantity  to  which  that  modulus  belongs  is  itself 
zero,  that  is, 


, 

(z  —  of 

so  that  /  (a)  =  —-.  !/-^~n  dz. 

ZTTI  )(z-  of 

This  theorem  evidently  corresponds  in  complex  variables  to  the  well-known 
theorem  of  differentiation  with  respect  to  a  constant  under  the  integral 
sign  when  all  the  quantities  concerned  are  real. 

Proceeding  in  the  same  way,  we  can  prove  that 

/  (a  +  &*)-/  (a)  _  2!_  f  /(*) 
Ba  ~2Trij(z-af 

where  6  is  a  small  quantity  which  vanishes  with  Ba.  Moreover  the  integral 
on  the  right-hand  side  is  finite,  for  the  subject  of  integration  is  everywhere 
finite  along  the  path  of  integration  which  itself  is  of  finite  length.  Hence, 
first,  a  small  change  in  the  independent  variable  leads  to  a  change  of  the 
same  order  of  small  quantities  in  the  value  of  the  function  f  (a),  which 
shews  that  f  (a)  is  a  continuous  function.  Secondly,  denoting 

&*)  -/(a) 


by  &/'(«),  we  have  the  limiting   value   of    -—  *—  -  equal  to  the  integral  on 

the  right-hand  side  when  Sa  vanishes,  that  is,  the  derivative  of  f  (a)  has 
a  value  independent  of  the  form  of  8a  and  therefore  /'  (a)  is  monogenic. 
Denoting  this  derivative  by  /"(a),  we  have 


J  (z  —  a)3 

Thirdly,  the  function  f  (a)   is  uniform  ;    for  it  is  the  limit  of  the  value 
of  —  -  --  x--  —  J-\J  and  both  /(a)  and  /(a  +  Sa)  are    uniform.      Lastly,  it 

is  finite;   for  (S  15)  it  is  the  value  of  the  integral  -  —  .  l.^—^dz,  in  which 

2?n  J  (z  —  af 

the  length  of  the  path  is  finite  and  the  subject  of  integration  is  finite  at 
every  point  of  the  path. 

Hence  f  (a)  is  continuous,  monogenic,  uniform,  and  finite  throughout 
the  whole  of  the  region  in  which  f  (z)  has  these  properties:  it  is  a 
holomorphic  function.  Hence  :  — 

When   a  function   is   holomorphic  in  any  region  of  the  plane   bounded 


32  PROPERTIES   OF  [21. 

by  a  simple  curve,  its  derivative  is  also  holomorphic  within  that  region.  And, 
by  repeated  application  of  this  theorem  : — 

When  a  function  is  holomorphic  in  any  region  of  the  plane  bounded 
by  a  simple  curve,  it  has  an  unlimited  number  of  successive  derivatives  each 
of  which  is- holomorphic  within  the  region. 

All  these  properties  have  been  shewn  to  depend  simply  upon  the  holo 
morphic  character  of  the  fundamental  function ;  but  the  inferences  relating 
to  the  derivatives  have  been  proved  only  for  points  within  the  region  and 
not  for  points  on  the  boundary.  If  the  foregoing  methods  be  used  to  prove 
them  for  points  on  the  boundary,  they  require  that  a  consecutive  point  shall 
be  taken  in  any  direction  ;  in  the  absence  of  knowledge  about  the  fundamental 
function  for  points  outside  (even  though  just  outside)  no  inferences  can  be 
justifiably  drawn. 

An  illustration  of  this  statement  is  furnished  by  the  hypergeometric  series 
which,  together  with  all  its  derivatives,  is  holomorphic  within  a  circle  of 
radius  unity  and  centre  the  origin ;  and  the  series  converges  unconditionally 
everywhere  on  the  circumference,  provided  7  >  a.  +  /3.  But  the  corresponding 
condition  for  convergence  on  the  circumference  ceases  to  be  satisfied  for  some 
one  of  the  derivatives  and  for  all  which  succeed  it :  as  such  functions  do  not 
then  converge  unconditionally,  the  circumference  of  the  circle  must  be 
excluded  from  the  region  within  which  the  derivatives  are  holomorphic. 

22.  Expressions  for  the  first  and  the  second  derivatives  have  been 
obtained. 

By  a  process  similar  to  that  which  gives  the  value  of  f  (a),  the  derivative 
of  order  n  is  obtainable  in  the  form 

n  '    f     f  (z\ 

/<»)  (a)  =  — .  I,          '      dz, 
J      w      2wt  J  (z  -  a)n+l 

the  integral  being  taken  round  the  whole  boundary  of  the  region  or  round  any 
curves  which  arise  from  deformation  of  the  boundary,  provided  that  no  point 
of  the  curves  in  the  final  or  any  intermediate  form  is  indefinitely  near  to  a. 

In  the  case  when  the  curve  of  integration  is  a  circle,  no  point  of  which 
circle  may  lie  outside  the  boundary  of  the  region,  we  have  a  modified  form 
fcr /*'(•> 

For  points  along  the  circumference  of  the  circle  with  centre  a  and  radius 

r,  let 

z  —  a  =  reei, 

dz 

so  that  as  before  —  =  idO : 

z  —  a 

then  0  and  2?r  being  taken  as  the  limits  of  0,  we  have 


22.]  HOLOMORPHIC    FUNCTIONS  33 

Let  M  be  the  greatest  value  of  the  modulus  of  f  (z)  for  points  on  the 
circumference  (or,  as  it  may  be  convenient  to  consider,  of  points  on  or  within 
the  circumference) :  then 


\f(n)(a)\<~  e-nei\\f(a 

i  /       \  / 1  ^  27ryw  *  - 

nl 

< 


M 


Now,  let  there  be  a  function  <£  (s)  defined  by  the  equation 

M 


—  a 


which  can  evidently  be  expanded  in  a  series  of  ascending  powers  of  z  —  a 
that  converges  within  the  circle.     The  series  is 


- 

[dnd>  (z)~\  ,  M 

Hence  —!L±J         =n\  — 

[    d*»    ]z=a      *>• 

so  that,  if  the  value  of  the  nth  derivative  of  $(z),  when  z  =  a,  be  denoted 
by  <£<n>  (a),  we  have 

|/»(a)|  «p>(a). 

These  results  can  be  extended  to  functions  of  more  than  one  variable  : 
the  proof  is  similar  to  the  foregoing  proof.  When  the  variables  are  two, 
say  z  and  z',  the  results  may  be  stated  as  follows  :  — 

^  For  all  points  z  within  a  given  simple  curve  0  in  the  ^-plane  and  all 
points  /  within  a  given  simple  curve  G'  in  the  /-plane,  let  /  (z,  z)  be  a 
holomorphic  function;  then,  if  a  be  any  point  within  C  and  a'  any  point 
within  G', 

^n+nJ  (a,  a') 


J  (z  -  a)n+1  (z'  —  aTf 

where  n  and  ri  are  any  integers  and  the  integral  is  taken  positively  round  the 
two  curves  G  and  G'. 

If  M  be  the  greatest  value  of  \f  (z,  z'}  for  points  z  and  z  within  their 
respective  regions  when  the  curves  G  and  G'  are  circles  of  radii  r,  r'  and 
centres  a,  a',  then 

dn+n'f(a,  a')  M 

~3aW»'        <w!/i!rv^5 

F. 


34  HOLOMORPHIC   FUNCTIONS  [22- 

M 
and  if  $(?>*) 


dn+n'f(a,a') 


dn+n'(j>  (z,  z') 


then  da»da'« 

when  z  =  a  and  z  =  a'  in  the  derivative  of  <£  (z,  z). 

23.  All  the  integrals  of  meromorphic  functions  that  have  been  considered 
have  been  taken  along  complete  curves :  it  is  necessary  to  refer  to  integrals 
along  curves  which  are  lines  only  from  one  point  to  another.  A  single 
illustration  will  suffice  at  present. 

Consider  the  integral  f  -t-^-dz;  the  function /»  is 

J  H0  z  —  a 

supposed  holomorphic  in  the  given  region,  and  z  and  z0  are 
any  two  points  in  that  region.  Let  some  curves  joining  z 
to  z0  be  drawn  as  in  the  figure  (fig.  7). 

~ ,  •*  2o 

is   holomorphic  over  the  whole  area  en-  Fig>  7 


z—  a 


closed  by  z^zSz0:  and  therefore  we  have  ^  =  0  when  taken  round  the 

boundary  of  that  area.     Hence  as  in  the  earlier  case  we  have 


z  —  a         Jz0  z  —  a 
The  point  a  lies  within  the  area  enclosed  by  z0yz^z0,  and  the  function 

is  holomorphic,  except  in  the  immediate  vicinity  of  z  =  a ;  hence 

r  f  (  v\ 

I  -          dz  =  2Trif(a), 

J  z  —  a 

the  integral  on  the  left-hand  side  being  taken  round  Z0yzj3z0.     Hence 


z  —  a 


Denoting  ^-by  g(z),  the  function  g  (z)  has  one  pole  a  in  the  region 

£  "~  CL 

considered. 

The  preceding  results  are  connected  only  with  the  simplest  form  of 
meromorphic  functions;  other  simple  results  can  be  derived  by  means  of  the 
other  theorems  proved  in  §§  17—21.  Those  which  have  been  obtained  are 
sufficient  however  to  shew  that  :  The  integral  of  a  meromorphic  function 
fg(z)dz  from  one  point  to  another  of  the  region  of  the  function  is  not  in 
general  a  uniform  function.  The  value  of  the  integral  is  not  altered  by 
any  deformation  of  the  path  which  does  not  meet  or  cross  a  pole  of  the 
function;  but  the  value  is  altered  when  the  path  of  integration  is  so 


23.]  GENERAL   PROPOSITIONS    IN   INTEGRATION  35 

deformed  as  to  pass  over  one  or  more  poles.  Therefore  it  is  necessary  to 
specify  the  path  of  integration  when  the  subject  of  integration  is  a  mero- 
morphic  function ;  only  partial  deformations  of  the  path  of  integration  are 
possible  without  modifying  the  value  of  the  integral. 

24.  The  following  additional  propositions*  are  deduced  from  limiting 
cases  of  integration  round  complete  curves.  In  the  first,  the  curve  becomes 
indefinitely  small ;  in  the  second,  it  becomes  infinitely  large.  And  in  neither, 
are  the  properties  of  the  functions  to  be  integrated  limited  as  in  the  pre 
ceding  propositions,  so  that  the  results  are  of  wider  application. 

I.  If  f(z)  be  a  function  which,  whatever  be  its  character  at  a,  has  no 
infinities  and  no  branch-points  in  the  immediate  vicinity  of  a,  the  value  of 
ff(z)dz  taken  round  a  small  circle  with  its  centre  at  a  tends  towards  zero 
when  the  circle  diminishes  in  magnitude  so  as  ultimately  to  be  merely  the 
point  a,  provided  that,  as  z  —  a  diminishes  indefinitely,  the  limit  of  (z  —  a)f(z) 
tend  uniformly  to  zero. 

Along  the  small  circle,  initially  taken  to  be  of  radius  r,  let 

*-a-fl*i  * 

dz 

so  that  =  idO, 

z—  a 

and  therefore  Sf(z)  dz  =  i\     (z  —  a)f(z)  d6. 

Jo 

Hence  \ff(z)dz\  =    I  *"  (z  -  a)f(z)  d0 

Jo 

<r\(z-a)f(z)\de 
Jo 

rzn 

<        Md0 
Jo 


where  M'  is  the  greatest  value  of  M,  the  modulus  of  (z  -  a)f(z),  for  points 
on  the  circumference.  Since  (z  -  a)f(z)  tends  uniformly  to  the  limit  zero  as 
|  z  -a  diminishes  indefinitely,  \jf(z)  dz\  is  ultimately  zero.  Hence  the  integral 
itself  jf(z)dz  is  zero,  under  the  assigned  conditions. 

Note.  If  the  integral  be  extended  over  only  part  of  the  circumference  of 
the  circle,  it  is  easy  to  see  that,  under  the  conditions  of  the  proposition, 
the  value  offf(z)de  still  tends  towards  zero. 

COROLLARY.  If  (z-a)f(z)  tend  uniformly  to  a  limit  k  as  \z-a\ 
diminishes  indefinitely,  the  value  of  ff(z)dz  taken  round  a  small  circle  centre 
a  tends  towards  27rik  in  the  limit. 

*  The  form  of  the  first  two  propositions,  which  is  adopted  here,  is  due  to  Jordan,  Cours 
d' Analyse,  t.  ii,  §§  285,  286. 

3—2 


36  GENERAL   PROPOSITIONS  [24. 

Thus  the  value  of   [-   dz    j,  taken  round  a  very  small  circle  centre  «,   where  a  is 

~  d  *  /2V 

not  the  origin,  is  zero  :  the  value  of  f  -  -  -  -,  round  the  same  circle  is  -.  (  -  \  . 

J  (a  —  z)  (a-M) 

Neither   the  theorem   nor   the   corollary  will  apply  to  a  function,  such  as  sn   —-^ 

which  has   the  point   a  for  an  essential  singularity:    the   value  of  (z-a)sn^—  ^,  as 
\z-a\  diminishes  indefinitely,  does  not  tend  (§  13)  to  a  uniform  limit.     As  a  matter  of 

fact   the  function  sn  —   has  an  infinite  number  of  poles  in  the  immediate  vicinity  of  a 
z-  a 

as  the  limit  z—a,  is  being  reached. 

II.  Whatever  be  the  character  of  a  function  f  (z}  for  infinitely  large  values 
ofz,  the  value  ofjf(z)  dz,  taken  round  a  circle  with  the  origin  for  centre,  tends 
towards  zero  as  the  circle  becomes  infinitely  large,  provided  that,  as  \z\ 
increases  indefinitely,  the  limit  of  zf(z)  tend  uniformly  to  zero. 

Along  a  circle,  centre  the  origin  and  radius  R,  we  have  z  =Eeei,  so  that 

dz      .ja 
-  =  idd, 

z 

r-2ir 

and  therefore  //  0)  dz  =  i      zf(z)  d6. 

Jo 

Hence  I  //(*)<&!  =  £* 

<T   zf(z)\dS 
Jo 

rzn 

<     Mde 

Jo 


< 

where  M'  is  the  greatest  value  of  M,  the  modulus  of  zf(z)t  for  points  on 
the  circumference.  When  R  increases  indefinitely,  the  value  of  M'  is  zero 
on  the  hypothesis  in  the  proposition;  hence  \$f(*)d*\  is  ultimately  zero. 
Therefore  the  value  of  ff(z)  dz  tends  towards  zero,  under  the  assigned  con 
ditions. 

Note.  If  the  integral  be  extended  along  only  a  portion  of  the  circumfer 
ence,  the  value  of  jf(z}dz  still  tends  towards  zero. 

COROLLARY.  //  zf(z)  tend  uniformly  to  a  limit  k  as  \z  .  increases 
indefinitely,  the  value  of  jf(z)  dz,  taken  round  a  very  large  circle,  centre  the 
origin,  tends  towards  %7rik. 

Thus  the  value  of  J(l  -zn}~^dz  round  an  infinitely  large  circle,  centre  the  origin,  is  zero 
if  n  >  2,  and  is  2ir  if  »  =  2. 

III.  If  all  the  infinities  and  the  branch-points  of  a  function  lie  in  a  finite 
region  of  the  z-plane,  then  the  value  of  jf(z)  dz  round  any  simple  curve,  which 


24.]  IN   INTEGRATION  37 

includes  all  those  points,  is  zero,  provided  the  value  of  zf(z\  as  \z\  increases 
indefinitely,  tends  uniformly  to  zero. 

The  simple  curve  can  be  deformed  continuously  into  the  infinite  circle 
of  the  preceding  proposition,  without  passing  over  any  infinity  or  any 
branch- point ;  hence,  if  we  assume  that  the  function  exists  all  over  the  plane, 
the  value  of  jf(z)  dz  is,  by  Cor.  I.  of  §  19,  equal  to  the  value  of  the  integral 
round  the  infinite  circle,  that  is,  by  the  preceding  proposition,  to  zero. 

Another  method  of  stating  the  proof  of  the  theorem  is  to  consider 
the  corresponding  simple  curve  on  Neumann's  sphere  (§  4).  The  surface 
of  the  sphere  is  divided  into  two  portions  by  the  curve*:  in  one  portion  lie 
all  the  singularities  and  the  branch-points,  and  in  the  other  portion  there  is 
no  critical  point  whatever.  Hence  in  this  second  portion  the  function  is  holo- 
morphic ;  since  the  area  is  bounded  by  the  curve  we  see  that,  on  passing  back 
to  the  plane,  the  excluded  area  is  one  over  which  the  function  is  holomorphic. 
Hence,  by  §  19,  the  integral  round  the  curve  is  equal  to  the  integral  round 
an  infinite  circle  having  its  centre  at  the  origin  and  is  therefore  zero,  as 
before. 

COROLLARY.  If,  under  the  same  circumstances,  the  value  of  zf(z},  as 
\z  increases  indefinitely,  tend  uniformly  to  k,  then  the  value  of  $f(z)dz  round 
the  simple  curve  is 


Thus  the  value  of  I  —      — r  along  any  simple  curve  which  encloses  the  two  points 
J  (a2  -  z2)* 

a  and  -  a  is  2ir ;  the  value  of 

dz 


{(!-«")  (!-*%•)}* 

round  any  simple  curve  enclosing  the  four  points  1,  -1,  T,  -7,  is  zero,  k  being  a  non- 

1C  K 

vanishing  constant ;  and  the  value  of  J(l  —  z2n)~*dz,  taken  round  a  circle,  centre  the  origin 
and  radius  greater  than  unity,  is  zero  when  n  is  an  integer  greater  than  1. 

/dz 
~  ~ — 771 

K*-«i)  (*-««)(*-«•)}* 

round  any  circle,  which  has  the  origin  for  centre  and  includes  the  three  distinct  points 
€lt  e2,  e3,  is  not  zero.  The  subject  of  integration  has  2  =  00  for  a  branch-point,  so  that  the 
condition  in  the  proposition  is  not  satisfied  ;  and  the  reason  that  the  result  is  no  longer 
valid  is  that  the  deformation  into  an  infinite  circle,  as  described  in  Cor.  I.  of  §  19, 
is  not  possible  because  the  infinite  circle  would  meet  the  branch-point  at  infinity. 

25.  The  further  consideration  of  integrals  of  functions,  that  do  not  possess 
the  character  of  uniformity  over  the  whole  area  included  by  the  curve  of  in 
tegration,  will  be  deferred  until  Chap.  ix.  Some  examples  of  the  theorems 
proved  in  the  present  chapter  will  now  be  given. 

*  The  fact  that  a  single  path  of  integration  is  the  boundary  of  two  portions  of  the  surface 
of  the  sphere,  within  which  the  function  may  have  different  characteristic  properties,  will  be 
used  hereafter  (§  104)  to  obtain  a  relation  between  the  two  integrals  that  arise  according  as  the 
path  is  deformed  within  one  portion  or  within  the  other. 


38  EXAMPLES    IN  [25. 

Ex.  1.  It  is  sufficient  merely  to  mention  the  indefinite  integrals  (that  is,  integrals  from 
an  arbitrary  point  to  a  point  z}  of  rational,  integral,  algebraical  functions.  After  the 
preceding  explanations  it  is  evident  that  they  follow  the  same  laws  as  integrals  of  similar 
functions  of  real  variables. 

/dz 
,— ^ ,  taken  round  a  simple  curve. 

When  n  is  0,  the  value  of  the  integral  is  zero  if  the  curve  do  not  include  the  point  a, 
and  it  is  Ziri  if  the  curve  include  the  point  a. 

When  n  is  a  positive  integer,  the  value  of  the  integral  is  zero  if  the  curve  do  not 
include  the  point  a  (by  §  17),  and  the  value  of  the  integral  is  still  zero  if  the  curve  do 
include  the  point  a  (by  §  22,  for  the  function  f(z)  of  the  text  is  1  and  all  its  derivatives 
are  zero).  Hence  the  value  of  the  integral  round  any  curve,  which  does  not  pass  through 
a,  is  zero. 

We  can  now  at  once  deduce,  by  §  20,  the  result  that,  if  a  holomorphic  function  be 
constant  along  any  simple  closed  curve  within  its  region,  it  is  constant  over  the  whole 
area  within  the  curve.  For  let  t  be  any  point  within  the  curve,  z  any  point  on  it,  and  C 
the  constant  value  of  the  function  for  all  the  points  z  ;  then 


B' 


mn 

2  —  t 

the  integral  being  taken  round  the  curve,  so  that 

<&M-—   t  dz 

=  C 
by  the  above  result,  since  the  point  t  lies  within  the  curve. 

Ex.  3.     Consider  the  integral  \e~^dz. 

In  any  finite  part  of  the  plane,  the  function  e~02  is  holomorphic;  therefore  (§  17)  the 
integral  round  the  boundary  of  a  rectangle 
(fig.  8),  bounded  by  the  lines  x=  ±a,  y  =  0, 
y=b,  is  zero :  and  this  boundary  can  be 
extended,  provided  the  deformation  remain 
in  the  region  where  the  function  is  holo 
morphic.  Now  as  a  tends  towards  infinity, 
the  modulus  of  e~z\  being  e~x2  +  y2,  tends 
towards  zero  when  y  remains  finite ;  and 
therefore  the  preceding  rectangle  can  be  Fig.  8. 

extended  towards  infinity  in  the  direction  of  the  axis  of  x,  the  side  b  of  the  rectangle 
remaining  unaltered. 

Along  A' A,  we  have  z=x  :  so  that  the  value  of  the  integral  along  the  part  A' A  of  the 

fa 

boundary  is  I      e~x  dx. 

J  -a 

Along  AB,  we  have  z  =  a  +  iy,  so  that  the  value  of  the  integral  along  the   part  AB 

f* 

is  i  I    e~(a  +  iyrdy. 

Jo 
Along  BB',  we  have  z  =  x  +  ib,  so  that  the  value  of  the  integral  along  the  part  BB' 

f'a 
is    I     e-(x  +  lVdx. 

J  a 

Along  B'A',  we  have  z=-a  +  iy,  so  that  the  value  of  the   integral   along  the  part 

B'A1  is  i  (V(-«H 
J  t, 


25.]  INTEGRATION  39 

/•ft 
The  second  of  these  portions  of  the  integral  is  e~a<i  .  »  .  I    tP~***tefy,  which  is  easily  seen 

J  o 
to  be  zero  when  the  (real)  quantity  a  is  infinite. 

Similarly  the  fourth  of  these  portions  is  zero. 

Hence  as  the  complete  integral  is  zero,  we  have,  on  passing  to  the  limit, 

I     e~^dx+\      e-^2ibx  +  b'2da;=0, 

J    -<*>  J  oo 

whence  e62  I      e~  *-***&?=*  I 

J  -oo  J  -<* 

/oo 
e'3^  (cos  2bx—i  sin 

and  therefore,  on  equating  real  parts,  we  obtain  the  well-known  result 


/ 

J     -Q 


This  is  only  one  of  numerous  examples*  in  which  the  theorems  in  the  text  can  be 
applied  to  obtain  the  values  of  definite  integrals  with  real  limits  and  real  variables. 

rzn-i 
Ex.  4.     Consider  the  integral  I  -  ---  dz.  where  n  is  a  real  positive  quantity  less  than 

J  1+z 
unity. 

The  only  infinities  of  the  subject  of  integration  are  the  origin  and  the  point  -  1  ; 
the  branch-points  are  the  origin  and  2=00.  Everywhere  else  in  the  plane  the  function 
behaves  like  a  holomorphic  function  ;  and,  therefore,  when  we  take  any  simple  closed 
curve  enclosing  neither  the  origin  nor  the  point  —  1,  the  integral  of  the  function  round 
that  curve  is  zero. 

We  shall  assume  that  the  curve  lies  on  the  positive  side  of  the  axis  of  x  and  that  it 
is  made  up  of  :  — 

(i)     a  semicircle  (73  (fig.  9),  centre  the  origin  and  radius  R  which  is  made  to  increase 
indefinitely  : 


Fig.  9. 


(ii)    two  semicircles,  ct  and  c2,  with  their  centres  at  0  and  —  1  respectively,  and  with 

radii  r  and  /,  which  ultimately  are  made  infinitesimally  small : 
(iii)     the  diameter  of  (73  along  the  axis  of  x  excepting  those  ultimately  infinitesimal 

portions  which  are  the  diameters  of  cx  and  of  c2. 

The  subject  of  integration  is  uniform  within  the  area  thus  enclosed  although  it 
is  not  uniform  over  the  whole  plane.  We  shall  take  that  value  of  zn~l  which  has  its 
argument  equal  to  (n—  1)  6,  where  6  is  the  argument  of  z. 

*  See  Briot  and  Bouquet,  Theorie  des  fonctions  elliptiques,  (2nd  ed.),  pp.  141  et  sqq.,  from 
which  examples  3  and  4  are  taken. 


40  EXAMPLES   IN  [25. 

The  integral  round  the  boundary  is  made  up  of  four  parts. 

0H  — 1 

(a)  The  integral  round  (73.     The  value  of  z .  ,  as  z  \  increases  indefinitely,  tends 

uniformly  to  the  limit  zero  ;   hence,  as  the  radius  of  the  semicircle  is  increased  indefinitely, 
the  integral  round  (73  vanishes  (§  24,  n.,  Note). 

^n— 1 

(b)  The  integral  round  cv     The  value  of  z  .   ,  as  |  z  \  diminishes  indefinitely, 

1  -\-z 

tends  uniformly  to  the  limit  zero ;   hence  as  the  radius  of  the  semicircle  is  diminished 
indefinitely,  the  integral  round  cv  vanishes  (§  24,  I.,  Note}. 

zn-l 

(c)  The  integral  round  c2.    The  value  of  (1  +  2) ,  as  |1+2|  diminishes  indefinitely 

A  ~r  z 

for  points  in  the  area,  tends  uniformly  to  the  limit  (—  I)""1,  i.e.,  to  the  limit  g(M~1)'™. 
Hence  this  part  of  the  integral  is 


being  taken  in  the  direction  indicated  by  the  arrow  round  c2)  the  infinitesimal  semicircle. 
Evidently  --       =id6  and  the  limits  are  TT  to  0,  so  that  this  part  of  the  whole  integral  is 

idd 


(d)    The  integral  along  the  axis  of  x.     The  parts  at  —  1  and  at  0  which  form  the 
diameters  of  the  small  semicircles  are  to  be  omitted  ;  so  that  the  value  is 


-l+r'      J  r 
This  is  what  Cauchy  calls  the  principal  value*  of  the  integral 

/"°°          /yH  ~  1 

/  •*  7 

I        dx. 

Since  the  whole  integral  is  zero,  we  have 

ineniri+  I      Y —  dx  =  0. 

Let  P  =  I     ^ —  dx,     P'  =  I       dx, 

and  0—  I dx, 

J  o  1-a? 

principal  values  being  taken  in  each  case.    Then,  taking  account  of  the  arguments,  we  have 


Since  iwenvi + P  + 1*  =  0, 

we  have  P  -  eH7riQ  =  -  inenni, 

*  Williamson's  Integral  Calculus,  %  104. 


25.]  INTEGRATION  41 

so  that  P—  Q  cos  nn  —  ir  sin  nn,     Q  sin  ntr  =  TT  cos  nir. 

Hence  I     ; dx—P  =  ir  cosec  TOTT, 

jo  1+a? 

dx—Q  =  ir  cot  %TT. 


.  5.     In  the  same  way  it  may  be  proved  that 


. 

where  n  is  an  integer,  a  is  positive  and  o>  is  e*2" . 

Jik  6.  By  considering  the  integral  Je-2^™-1^  round  the  contour  of  the  sector  of  a 
circle  of  radius  r,  bounded  by  the  radii  0=0,  6=a,  where  a  is  less  than  |TT  and  n  is  positive, 
it  may  be  proved  that 


„»— 1  „-; 


{r'; 
on  proceeding  to  the  limit  when  r  is  made  infinite.     (Briot  and  Bouquet.) 

Ex.  7.     Consider  the  integral  I  ~~^,  where  n  is  an  integer.     The  subject  of  integration 

is  meromorphic  ;  it  has  for  its  poles  (each  of  which  is  simple)  the  n  points  o>r  for  r=0, 
1,  ...,  n-l,  where  a  is  a  primitive  nth  root  of  unity ;  and  it  has  no  other  infinities  and  no 

branch -points.     Moreover  the  value  of  — — -,  as  \z\  increases  indefinitely,  tends  uniformly 

to  the  limit  zero  ;  hence  (§  24,  in.)  the  value  of  the  integral,  taken  round  a  circle  centre 
the  origin  and  radius  >  1,  is  zero. 

This  result  can  be  derived  by  means  of  Corollary  II.  in  §  19.  Surround  each  of  the 
poles  with  an  infinitesimal  circle  having  the  pole  for  centre ;  then  the  integral  round 
the  circle  of  radius  >  1  is  equal  to  the  sum  of  the  values  of  the  integral  round  the 
infinitesimal  circles.  The  value  round  the  circle  having  «r  for  its  centre  is,  by  §  20, 


2rri(  limit  of       "  ,  when  z  =  u>r} 
\  z  -  L  J 


Hence  the  integral  round  the  large  circle 


2 

n    r=n 


=  0. 


Ex.  8.  Hitherto,  in  all  the  examples  considered,  the  poles  that  have  occurred  have 
been  simple :  but  the  results  proved  in  §  21  enable  us  to  obtain  the  integrals  of 
functions  which  have  multiple  poles  within  an  area.  As  an  example,  consider  the 

integral  /  (1+g2)n  +  i  round  any  curve  which  includes  the  point  i  but  not  the  point  -  i,  these 
points  being  the  two  poles  of  the  subject  of  integration,  each  of  multiplicity  n  +  l. 


42  EXAMPLES   IN    INTEGRATION  [25. 

We  have  seen  that  /"  (a)  =  j^  J  ^_a^n  +  i *» 

where  /(«)  is  holomorphic  throughout  the  region  bounded  by  the  curve  round  which  the 
integral  is  taken. 

In  the  present  case  a  is  i,  and  f(z)  = .       «.n ^\  ',  s°  that 

2» !     (-l)n 


u.     <•  /*n  -  2- 

and  therefore  /"  (*J  =  ^j  (2i)»*-n  "~  ~  wT 

Hence  we  have  "***™' 


In  the  case  of  the  integral  of  a  function  round  a  simple  curve  which  contains  several 
of  its  poles  we  first  (§  20)  resolve  the  integral  into  the  sum  of  the  integrals  round  simple 
curves  each  containing  only  one  of  the  points,  and  then  determine  each  of  the  latter 
integrals  as  above. 

Another  method  that  is  sometimes  possible  makes  use  of  the  expression  of  the  uniform 
function  in  partial  fractions.  After  Ex.  2,  we  need  retain  only  those  fractions  which  are  of 

the  form  —  :  the  integral  of  such  a  fraction  is  ZniA,  and  the  value  of  the  whole  integral 

z-a 

is  therefore  tor&A.  It  is  thus  sufficient  to  obtain  the  coefficients  of  the  inverse  first  powers 
which  arise  when  the  function  is  expressed  in  partial  fractions  corresponding  to  each  pole. 
Such  a  coefficient  A,  the  coefficient  of  -j  in  the  expansion  of  the  function,  is  called  by 

Z       (Jj 

Cauchy  the  residue  of  the  function  relative  to  the  point. 
For  example, 


so  that  the  residues  relative  to  the  points  -1,  -o>,  -to2  are  f,  £«,  |«2  respectively. 
Hence  if  we  take  a  semicircle,  of  radius  >  1  and  centre  the  origin  with  its  diameter 
along  the  axis  of  y,  so  as  to  lie  on  the  positive  side  of  the  axis  of  y,  the  area  between  the 
semi-circumference  and  the  diameter  includes  the  two  points  -«  and  -«2  ;  and  therefore 

the  value  of 

dz 


taken  along  the  semi-circumference  and  the  diameter,  is 

&*&»+!•?); 

i.e.,  the  value  is  -  *ni. 


CHAPTER   III. 

EXPANSION  OF  FUNCTIONS  IN  SERIES  OF  POWERS. 

26.  WE  are  now  in  a  position  to  obtain  the  two  fundamental  theorems 
relating  to  the  expansion  of  functions  in  series  of  powers  of  the  variable : 
they  are  due  to  Cauchy  and  Laurent  respectively. 

Cauchy 's  theorem  is  as  follows*: — 

When  a  function  is  holomorphic  over  the  area  of  a  circle  of  centre  a,  it  can 
be  expanded  as  a  series  of  positive  integral  powers  of  z-a  converging  for  all 
points  within  the  circle. 

Let  z  be  any  point  within  the  circle;  describe  a  concentric  circle  of 
radius  r  such  that 

\z-a\  =  p  <r<R,  ^ ^i. 

where  R  is  the  radius  of  the  given  circle.  If  t 
denote  a  current  point  on  the  circumference  of  the 
new  circle,  we  have 


dt 


t  —  a        z  —  a 
t  —  a 


Fif?.   10. 


the  integral  extending  along  the  whole  circumference  of  radius  r.     Now 


z-a 


t-a 


z  -an+l 


z  —  a 


—  a 


t—a 


so  that,  by  §  14  (III.),  we  have 


J_  f 
27ri] 


f(t) 


t-z\t-a 


dt. 


*  Exercices  d' 'Analyse  et  de  Physique  Mathe'matiqne,  t.  ii,  pp.  50  et  seq. ;  the  memoir  was  first 
made  public  at  Turin  in  1832. 


44 


CAUCHY'S  THEOREM  ON  THE  [26. 

Now  /(«)  is  holomorphic  over  the  whole  area  of  the  circle  ;  hence,  if  t  be 
not  actually  on  the  boundary  of  the  region  (§§  21,  22),  a  condition  secured  by 
the  hypothesis  r  <  R,  we  have 


and  therefore 

(z-a)n  (z-a)n+l 

" 


Let  the  last  term  be  denoted  by  L.  Since  z  —  a  =p  and  \t-a\  =  r, 
it  is  at  once  evident  that  \t-z\^r-p.  Let  M  be  the  greatest  value  of 
|/(0|  for  points  along  the  circle  of  radius  r ;  then  M  must  be  finite,  owing  to 
the  initial  hypothesis  relating  tof(z).  Taking 

f  —  n  —  TP6i 

v  W/  ~~  I  C* 

so  that  dt  =  i(t-  a)  d6, 

P«+>  t*m    de 

we  have  \L\  =  -f 

i 

^P 


Jlf 


rn  (i —  p) 

Jffl-:C 


\r) 
Now  r  was  chosen  to  be  greater  than  p  ;  hence  as  n  becomes  infinitely 

large,    we    have    W       infinitesimally  small.     Also  If  (1  —  ?l       is   finite. 
\r/  V        f/ 

Hence  as  ?i  increases  indefinitely,  the  limit  of  |i|,  necessarily  not  negative, 
is  infinitesimally  small  and  therefore,  in  the  same  case,  L  tends  towards 
zero. 

It  thus  appears,  exactly  as  in  §  15  (V.),  that,  when  n  is  made  to  increase 
without  limit,  the  difference  between  the  quantity  f(z)  and  the  first  n  +  1 
terms  of  the  series  is  ultimately  zero  ;  hence  the  series  is  a  converging  series 
having  f(z)  as  the  limit  of  the  sum,  so  that 


which  proves  the  proposition  under  the  assigned  conditions.     It  is  the  form 
of  Taylor's  expansion  for  complex  variables. 

Note.  The  series  on  the  right-hand  side  is  frequently  denoted  by 
P(z  —  a),  where  P  is  a  general  symbol  for  a  converging  series  of  positive 
integral  powers  of  z  —  a:  it  is  also  sometimes*  denoted  by  P(z\a).  Con- 

*  Weierstrass,  Abh.  am  der  Functionenlehre,  p.  1. 


26.]  EXPANSION   OF   A    FUNCTION  45 

formably  with   this  notation,  a  series  of  negative  integral  powers  of  z  —  a 

would  be  denoted  by  P I  -    — ) ;  a  series  of  negative  integral  powers  of  z 

\z  —  a/ 

either  by  P  (-)  or  by  P(^|oo),  the  latter  implying  a  series  proceeding  in 
\zj 

positive  integral  powers  of  a  quantity  which  vanishes  when  z  is  infinite, 
i.e.,  in  positive  integral  powers  of  — . 

Z 

If,  however,  the  circle  can  be  made  of  infinitely  great  radius  so  that  the 
function  f(z)  is  holomorphic  over  the  finite  part  of  the  plane,  the  equivalent 
series  is  denoted  by  G(z  —  a)  and  it  converges  over  the  whole  plane. 
Conformably  with  this  notation,  a  series  of  negative  integral  powers  of  z  -  a 

which  converges  over  the  whole  plane  is  denoted  by  G  I  -    -  j . 

27.  The  following  remarks  on  the  proof  and  on  inferences  from  it  should 
be  noticed. 

(i)     In  order  that  — -  -  may  be  expanded  in  the  required  form,  the 
t  —  z 

point  z  must  be  taken  actually  within  the  area  of  the  circle  of  radius  R ; 
and  therefore  the  convergence  of  the  series  P  (z  —  a)  is  not  established  for 
points  on  the  circumference. 

(ii)  The  coefficients  of  the  powers  of  z  —  a  in  the  series  are  the 
values  of  the  function  and  its  derivatives  at  the  centre  of  the  circle ;  and  the 
character  of  the  derivatives  is  sufficiently  ensured  (§  21)  by  the  holomorphic 
character  of  the  function  for  all  points  within  the  region.  It  therefore 
follows  that,  if  a  function  be  holomorphic  within  a  region  bounded  by  a 
circle  of  centre  a,  its  expansion  in  a  series  of  ascending  powers  of  z  —  a 
converging  for  all  points  within  the  circle  depends  only  upon  the  values  of 
the  function  and  its  derivatives  at  the  centre. 

But  instead  of  having  the  values  of  the  function  and  of  all  its  derivatives 
at  the  centre  of  the  circle,  it  will  suffice  to  have  the  values  of  the  holomorphic 
function  itself  over  any  small  region  at  a  or  along  any  small  line  through 
a,  the  region  or  the  line  not  being  infinitesimal.  The  values  of  the 
derivatives  at  a  can  be  found  in  either  case ;  for  /'  (b)  is  the  limit  of 
{f(b  +  86)  —f(b)}/8b,  so  that  the  value  of  the  first  derivative  can  be  found 
for  any  point  in  the  region  or  on  the  line,  as  the  case  may  be ;  and  so  for  all 
the  derivatives  in  succession. 

(iii)  The  form  of  Maclaurin's  series  for  complex  variables  is  at  once 
derivable  by  supposing  the  centre  of  the  circle  at  the  origin.  We  then 
infer  that,  if  a  function  be  kolomorphic  over  a  circle,  centre  tJie  origin,  it  can  be 


46  DARBOUX'S    EXPRESSION  [27. 

represented  in  the  form  of  a  series  of  ascending,  positive,  integral  powers  of  the 
variable  given  by 


where  the  coefficients  of  the  various  powers  of  z  are  the  values  of  the  derivatives 
of f(z)  at  the  origin,  and  the  series  converges  for  all  points  within  the  circle. 

Thus,  the  function  ez  is  holomorphic  over  the  finite  part  of  the  plane ; 
therefore  its  expansion  is  of  the  form  G  (z).  The  function  log  (1  4-  z)  has  a 
singularity  at  —  1 ;  hence  within  a  circle,  centre  the  origin  and  radius  unity, 
it  can  be  expanded  in  the  form  of  an  ascending  series  of  positive  integral 
powers  of  z,  it  being  convenient  to  choose  that  one  of  the  values  of  the 
function  which  is  zero  at  the  origin.  Again,  tan"1.?2  has  singularities  at  the 
four  points  z4  =  —  I,  which  all  lie  on  the  circumference;  choosing  the  value  at 
the  origin  which  is  zero  there,  we  have  a  similar  expansion  in  a  series,  con 
verging  for  points  within  the  circle. 

Similarly  for  the  function  (1  +z)n,  which  has  —  1  for  a  singularity. 

(iv)  Darboux's  method*  of  derivation  of  the  expansion  of  f  (z)  in 
positive  powers  of  z  —  a  depends  upon  the  expression,  obtained  in  §  15  (IV.), 
for  the  value  of  an  integral.  When  applied  to  the  general  term 

1     Uz-a\n+i  s,..  ,, 
f(t)dt, 


=  L  say,  it  gives  L  =  \r  fe^J      /(f), 

where  £  is  some  point  on  the  circumference  of  the  circle  of  radius  r,  and  X  is 

2     ~    fl 

a  complex  quantity  of  modulus  not  greater  than  unity.   The  modulus  of  ^ 

b  ~~  a 

is  less  than  a  quantity  which  is  less  than  unity ;  the  terms  of  the  series  of 
moduli  are  therefore  less  than  the  terms  of  a  converging  geometric  progres 
sion,  so  that  they  form  a  converging  series;  the  limit  of  \L\,  and  therefore 
of  L,  can,  with  indefinite  increase  of  n,  be  made  zero  and  Taylor's  expansion 
can  be  derived  as  before. 

00 

Ex.  1.     Prove  that  the  arithmetic  mean  of  all  values  of  z~  n  2  avzv,  for  points  lying  along 

v  =  0 

a  circle  |z|  =  r  entirely  contained  in  the  region  of  continuity,  is  an.     (Rouche,  Gutzmer.) 
Prove  also  that  the  arithmetic  mean  of  the  squares  of  the  moduli  of  all  values  of 

00 

2  avzv,  for  points  lying  along  a  circle  z\  =  r  entirely  contained  in  the  region  of  continuity, 

x  =  0 

is  equal  to  the  sum  of  the  squares  of  the  moduli  of  the  terms  of  the  series  for  a  point  on 
the  circle.  (Gutzmer.) 

00 

Ex.  2.     Prove  that  the  function  2  anzn*, 

M  =  0 

is  finite  and  continuous,  as  well  as  all  its  derivatives,  within  and  on  the  boundary  of  the 
circle  |0|  =  1,  provided  a  <  1.  (Fredholm.) 

*  Liouville,  3dmc  Ser.,  t.  ii,  (1876),  pp.  291—312. 


28.] 


LAURENT'S  EXPANSION  OF  A  FUNCTION 


47 


28.     Laurent's  theorem  is  as  follows*: — 

A  function,  which  is  holomorphic  in  a  part  of  the  plane  bounded  by  two 
concentric  circles  with  centre  a  and  finite  radii,  can  be  expanded  in  the  form 
of  a  double  series  of  integral  powers,  positive  and  negative,  of  z  —  a,  the  series 
converging  uniformly  and  unconditionally  in  the  part  of  the  plane  between  the 
circles. 

Let  z  be  any  point  within  the  region  bounded  by  the  two  circles  of  radii 
R   and  R;    describe    two    concentric    circles    of 
radii  r  arid  r'  such  that 

R>r>  z-a  >r'>  R. 

Denoting  by  t  and  by  s  current  points  on  the 
circumference  of  the  outer  and  of  the  inner 
circles  respectively,  and  considering  the  space 
which  lies  between  them  and  includes  the  point 
z,  we  have,  by  §  20, 

/w-oL 


:  —  Z  ZTTlJs  —  2"~  Fig.   11. 

a  negative  sign  being  prefixed  to  the  second  integral  because  the  direction 
indicated  in  the  figure  is  the  negative  direction  for  the  description  of  the 
inner  circle  regarded  as  a  portion  of  the  boundary. 

Now  we  have 

fz  —  a" 
t  —  a      _      z  — 

t 


'  —  a      Iz  —  a\* 
—  a      \t-aj 


z-  a. 

+  I  .  —  -      + 
—  a. 


1  - 


z  —  a 


t  —  a 

this  expansion  being  adopted  with  a  view  to  an  infinite  converging  series, 
z  —  a 


because 


t  —  a 


is  less  than  unity  for  all  points  t;   and  hence,  by  §  15, 


_  n\n+l 


dt. 


—  z  \t  —  a/ 

Now  each  of  the  integrals,  which  are  the  respective  coefficients  of  powers  of 
z  —  a,  is  finite,  because  the  subject  of  integration  is  everywhere  finite  along 
the  circle  of  finite  radius,  by  §  15  (IV.).  Let  the  value  of 

^r*      %  '•'-'••- 

be  2iriur :  the  quantity  ur  is  not  necessarily  equal  to  /''  (a)  -r-  r  I,  because  no 
*  Comptes  Rendus,  t.  xvii,  (1843),  p.  939. 


48  LAURENT'S  EXPANSION  OF  [28. 

knowledge  of  the  function  or  of  its  derivatives  is  given  for  a  point  within 
the  innermost  circle  of  radius  R'.     Thus 

_L  f/2)  dt  =  u0  +  (z -  a) u1  +  (z-  a)2 w2+ +(z- a)nun 

2w»  J  t  —  z 

1     [f  (t)  (z  —  a\n+1  -, 


-  z  \t  —  a 

The  modulus  of  the  last  term  is  less  than 

M 


where  p  is  z-a  and  If  is  the  greatest  value  of  \f(t)\  for  points  along  the 
circle.  Because  p  <  r,  this  quantity  diminishes  to  zero  with  indefinite  in 
crease  of  n  ;  and  therefore  the  modulus  of  the  expression 


v        % 

becomes  indefinitely  small  with  increase  of  n.  The  quantity  itself  therefore 
vanishes  in  the  same  limiting  circumstance  ;  and  hence 

1  .  [fl&dt  =  u0  +  (z-<i)u1  +  ......  +(z-a)mum+  ......  , 

2-7TI  J  t  —  Z 

so  that  the  first  of  the  integrals  is  equal  to  a  series  of  positive  powers.  This 
series  converges  uniformly  and  unconditionally  within  the  outer  circle,  for 
the  modulus  of  the  (m  +  l)th  term  is  less  than 


which  is  the  (m  +  l)th  term  of  a  converging  series*. 

As  in  §  27,  the  equivalence  of  the  integral  and  the  series  can  be  affirmed 
only  for  points  which  lie  within  the  outermost  circle  of  radius  R. 

Again,  we  have 

fs  -  a\n+1 


z-a  _         s-a  fs  -  a\n      (z-a) 


s-z  z-a  \z-a 

z  —  a 

this  expansion  being  adopted  with  a  view  to  an  infinite  converging  series, 


because 


s  —  a 


z  —  a 


is  less  than  unity.     Hence 


1     [/s-a\ 
.  If  -    - 
2?rt  J  \z-aj 


-n+1f(s) 
J-~- 


, 

-ds. 
z  —  s 


Chrystal,  ii,  124. 


28.]  A   FUNCTION   IN    SERIES  49 

The  modulus  of  the  last  term  is  less  than 

M' 


P 

where  M'  is  the  greatest  value  of  \f(s)\  for  points  along  the  circle  of  radius 
r'.  With  indefinite  increase  of  n,  this  modulus  is  ultimately  zero  ;  and  thus, 
by  an  argument  similar  to  the  one  which  was  applied  to  the  former  integral, 
we  have 


..  ..    -          .. 

ZTTI  J  s  —  z          z  —  a     (z  —  a)2  (z  —  a)m 

where  vm  denotes  the  integral  f(s  —  a)m~lf  (s)  ds  taken  round  the  circle. 

As  in  the  former  case,  the  series  is  one  which  converges  uniformly  and 
unconditionally;  and  the  equivalence  of  the  integral  and  the  series  is  valid 
for  points  z  that  lie  without  the  innermost  circle  of  radius  R'. 

The  coefficients  of  the  various  negative  powers  of  z  —  a  are  of  the  form 

1     f   /(*)     d(    1    ^ 
tori]  __  1_        (s-a)' 

(s  -  a)m 
a  form  that  suggests  values  of  the  derivatives  of  f  (s)  at  the  point  given  by 

-  =  0,  that  is,  at  infinity.  But  the  outermost  circle  is  of  finite  radius  ; 
s-a 

and  no  knowledge  of  the  function  at  infinity,  lying  without  the  circle,  is 
given,  so  that  the  coefficients  of  the  negative  powers  may  not  be  assumed 
to  be  the  values  of  the  derivatives  at  infinity,  just  as,  in  the  former  case,  the 
coefficients  ur  could  not  be  assumed  to  be  the  values  of  the  derivatives  at  the 
common  centres  of  the  circles. 

Combining  the  expressions  obtained  for  the  two  integrals,  we  have 
f(z)  =  u0  +  (z  —  a)  u-i  +  (z  —  a)2  w2  +  ... 

+  (z-  a)-1  Vl  +  (z-  a)~2  va+  .... 

Both  parts  of  the  double  series  converge  uniformly  and  unconditionally  for 
all  points  in  the  region  between  the  two  circles,  though  not  necessarily  for 
points  on  the  boundary  of  the  region.  The  whole  series  therefore  converges 
for  all  those  points  :  and  we  infer  the  theorem  as  enunciated. 

Conformably  with  the  notation  (§  26,  note)  adopted  to  represent  Taylor's 
expansion,  a  function  f(z)  of  the  character  required  by  Laurent's  Theorem 
can  be  represented  in  the  form 


the  series  P1  converging  within  the  outer  circle  and  the  series  P2  converging 
without  the  inner  circle  ;  their  sum  converges  for  the  ring-space  between  the 
circles. 

F.  4 


50  LAURENT'S  THEOREM  [29. 

29.     The  coefficient  u0  in  the  foregoing  expansion  is 

-1-  f  £9  dt 

torijt-a     ' 

the  integral  being  taken  round  the  circle  of  radius  r.     We  have 

dt  =ide 


t  —  a 
for  points  on  the  circle  ;  and  therefore 

d0 


so  that  \u0\<!deMt<M', 

J  ZTT 

M'  being  the  greatest  value  of  Mt,  the  modulus  of  f(t),  for  points  along  the 
circle.  If  M  be  the  greatest  value  of  \f(z}\  for  any  point  in  the  whole 
region  in  which  f(z)  is  defined,  so  that  M'^.M,  then  we  have 

«o  1  <  M, 

that  is,  the  modulus  of  the  term  independent  of  z  —  a  in  the  expansion  of 
f(z)  by  Laurent's  Theorem  is  less  than  the  greatest  value  of  \f(z)  \  at  points 
in  the  region  in  which  it  is  defined. 

Again,  (z-a)-mf(z)  is  a  double  series  in  positive  and  negative  powers  of 
z-a,  the  term  independent  of  z  -a  being  um;  hence,  by  what  has  just  been 
proved,  um  \  is  less  than  p~m  M,  where  p  is  z  -  a  .  But  the  coefficient  um 
does  not  involve  z,  and  we  can  therefore  choose  a  limit  for  any  point  z.  The 
lowest  limit  will  evidently  be  given  by  taking  z  on  the  outer  circle  of  radius 
R,  so  that  um  <  MR~m.  Similarly  for  the  coefficients  vm  ;  and  therefore  we 
have  the  result  :  — 

If  f(z)  be  expanded  as  by  Laurent's  Theorem  in  the  form 

OO  00 

u0+   2  (z-a)mum+  2   (z-aY^Vm, 

m  =  l  m=l 

then  \um  <MR~m,     \vm  <MR'm, 

where  M  is  the  greatest  value  of  \f(z)    at  points  within  the  region  in  which 

f(z)  is  defined,  and  R  and  R'  are  the  radii  of  the  outer  and  the  inner  circles 

respectively. 

30.  The  following  proposition  is  practically  a  corollary  from  Laurent's 
Theorem  :  — 

When  a  function  is  holomorphic  over  all  the  plane  which  lies  outside  a 
circle  of  centre  a,  it  can  be  expanded  in  the  form  of  a  series  of  negative  integral 
powers  of  z  —  a,  the  series  converging  uniformly  and  unconditionally  everywhere 
in  that  part  of  the  plane. 

It  can  be  deduced  as  the  limiting  case  of  Laurent's  Theorem  when  the 


30.]  EXPANSION   IN   NEGATIVE   POWERS  51 

radius  of  the  outer  circle  is  made  infinite.     We  then  take  r  infinitely  large, 
and  substitute  for  t  by  the  relation 

t  —  a  =  reei, 

so  that  the  first  integral  in  the  expression  (a),  p.  47,  for/(^)  is 

1    f2"    d0 


t  —  a 

Since  the  function  is  holomorphic  over  the  whole  of  the  plane  which  lies 
outside  the  assigned  circle,  f(t}  cannot  be  infinite  at  the  circle  of  radius  r 
when  that  radius  increases  indefinitely.  If  it  tend  towards  a  (finite)  limit  k, 
which  must  be  uniform  owing  to  the  hypothesis  as  to  the  functional  character 
of  f(z\  then,  since  the  limit  of  (t  —  z)/(t  —  a)  is  unity,  the  preceding  integral 
is  equal  to  k. 

The  second  integral  in  the  same  expression  (a),  p.  47,  for  f(z)  is  un 
altered  by  the  conditions  of  the  present  proposition ;  hence  we  have 

f(z)  =  k  +  (z-  a)~l  vl  +  (z-  a)-2Vz  +  ..., 

the  series  converging  uniformly  and  unconditionally  without  the  circle, 
though  it  does  not  necessarily  converge  on  the  circumference. 

The  series  can  be  represented  in  the  form 

1 


\z  —  a/ 
conformably  with  the  notation  of  §  26. 

Of  the  three  theorems  in  expansion  which  have  been  obtained,  Cauchy's 
is  the  most  definite,  because  the  coefficients  of  the  powers  are  explicitly 
obtained  as  values  of  the  function  and  of  its  derivatives  at  an  assigned  point. 
In  Laurent's  theorem,  the  coefficients  are  not  evaluated  into  simple  expres 
sions  ;  and  in  the  corollary  frofti  Laurent's  theorem  the  coefficients  are,  as  is 
easily  proved,  the  values  of  the  function  and  of  its  derivatives  for  infinite 
values  of  the  variable.  The  essentially  important  feature  of  all  the  theorems 
is  the  expansibility  of  the  function  in  series  under  assigned  conditions. 

31.  It  was  proved  (§21)  that,  when  a  function  is  holomorphic  in  any 
region  of  the  plane  bounded  by  a  simple  curve,  it  has  an  unlimited  number 
of  successive  derivatives  each  of  which  is  holomorphic  in  the  region.  Hence, 
by  the  preceding  propositions,  each  such  derivative  can  be  expanded  in 
converging  series  of  integral  powers,  the  series  themselves  being  deducible 
by  differentiation  from  the  series  which  represents  the  function  in  the  region. 

In  particular,  when  the  region  is  a  finite  circle  of  centre  a,  within  which 
f(z)  and  consequently  all  the  derivatives  off(z)  are  expansible  in  converging 
series  of  positive  integral  powers  of  z  —  a,  the  coefficients  of  the  various 
powers  of  z  —  a  are — save  as  to  numerical  factors — the  values  of  the 

4—2 


52  DEFINITION   OF   DOMAIN  [31. 

derivatives  at  the  centre  of  the  circle.  Hence  it  appears  that,  when  a  function 
is  holomorphic  over  the  area  of  a  given  circle,  the  values  of  the  function  and  all 
its  derivatives  at  any  point  z  within  the  circle  depend  only  upon  the  variable 
of  the  point  and  upon  the  values  of  the  function  and  its  derivatives  at  the 
centre. 

32.  Some  of  the  classes  of  points  in  a  plane  that  usually  arise  in 
connection  with  uniform  functions  may  now  be  considered. 

(i)  A  point  a  in  the  plane  may  be  such  that  a  function  of  the  variable 
has  a  determinate  finite  value  there,  always  independent  of  the  path  by 
which  the  variable  reaches  a ;  the  point  a,  is  called  an  ordinary  point*  of  the 
function.  The  function,  supposed  continuous  in  the  vicinity  of  a,  is  con 
tinuous  at  a :  and  it  is  said  to  behave  regularly  in  the  vicinity  of  an  ordinary 
point. 

Let  such  an  ordinary  point  a  be  at  a  distance  d,  not  infinitesimal,  from 
the  nearest  of  the  singular  points  (if  any)  of  the  function ;  and  let  a  circle  of 
centre  a  and  radius  just  less  than  d  be  drawn.  The  part  of  the  z-plane  lying 
within  this  circle  is  calledf  the  domain  of  a ;  and  the  function,  holomorphic 
within  this  circle,  is  said  to  behave  regularly  (or  to  be  regular)  in  the  domain 
of  a.  From  the  preceding  section,  we  infer  that  a  function  and  its  derivatives 
can  be  expanded  in  a  converging  series  of  positive  integral  powers  of  z  —  a 
for  all  points  z  in  the  domain  of  a,  an  ordinary  point  of  the  function :  and 
the  coefficients  in  the  series  are  the  values  of  the  function  and  its  derivatives 
at  a. 

The  property  possessed  by  the  series — that  it  contains  only  positive 
integral  powers  of  z  -  a— at  once  gives  a  test  that  is  both  necessary  and 
sufficient  to  determine  whether  a  point  is  an  ordinary  point.  If  the  point  a 
be  ordinary,  the  limit  of  (z  -  a)  f  (z}  necessarily  is  zero  when  z  becomes  equal 
to  a.  This  necessary  condition  is  also  sufficient  to  ensure  that  the  point  is 
an  ordinary  point  of  the  function  /  (z),  supposed  to  be  uniform ;  for,  since 
f(z)  is  holomorphic,  the  function  (z-a)f(z)  is  also  holomorphic  and  can  be 
expanded  in  a  series 

M0  -f  wa  (z  —  d)  +  w2  (?  —  a)2  +  •  •  -, 

converging  in  the  domain  of  a.  The  quantity  u0  is  zero,  being  the  value 
of  (z-a)f(z)  at  a  and  this  vanishes  by  hypothesis;  hence 

(z-a)f  (z)  =  (z  —  a)  {MI  +  u2(z -a) +...}, 

shewing  that  /  (z)  is  expressible  as  a  series  of  positive  integral  powers  of 
z—  a  converging  within  the  domain  of  a,  or,  in  other  words,  that/(*)  certainly 
has  a  for  an  ordinary  point  in  consequence  of  the  condition  being  satisfied. 

*  Sometimes  a  regular  point. 

t  The  German  title  is  Umgebung,  the  French  is  domaine. 


32.]  ESSENTIAL   SINGULARITY  53 

(ii)  A  point  a  in  the  plane  may  be  such  that  a  function  /  (z)  of  the 
variable  has  a  determinate  infinite  value  there,  always  independent  of  the 
path  by  which  the  variable  reaches  a,  the  function  behaving  regularly  for 

points  in  the  vicinity  of  a ;  then  ^—\  nas  a  determinate  zero  value  there,  so 

/  (?) 

that  a  is  an  ordinary  point  of   --r-r  .     The  point  a  is  called  a  pole  (§12)  or 
an  accidental  singularity*  of  the  function. 

A  test,  necessary  and  sufficient  to  settle  whether  a  point  is  an  accidental 
singularity  of  a  function  will  subsequently  (§  42)  be  given. 

(iii)  A  point  a  in  the  plane  may  be  such  that  y (2)  has  not  a  determinate 
value  there,  either  finite  or  infinite,  though  the  function  is  regular  for  all 

points  in  the  vicinity  of  a  that  are  not  at  merely  infinitesimal  distances. 

i        1 
Thus  the  origin  is  of  this  nature  for  the  functions  ez,  sn  - . 

Z 

Such  a  point  is  called-f*  an  essential  singularity  of  the  function.  No 
hypothesis  is  postulated  as  to  the  character  of  the  function  for  points 
at  infinitesimal  distances  from  the  essential  singularity,  while  the  relation 
of  the  singularity  to  the  function  naturally  depends  upon  this  character  at 
points  near  it.  There  may  thus  be  various  kinds  of  essential  singularities 
all  included  under  the  foregoing  definition ;  their  classification  is  effected 
through  the  consideration  of  the  character  of  the  function  at  points  in  their 
immediate  vicinity.  (See  §  88.) 

One  sufficient  test  of  discrimination  between  an  accidental  singularity 
and  an  essential  singularity  is  furnished  by  the  determinateness  of  the  value 
at  the  point.  If  the  reciprocal  of  the  function  have  the  point  for  an  ordinary 
point,  the  point  is  an  accidental  singularity — it  is,  indeed,  a  zero  for  the 
reciprocal.  But  when  the  point  is  an  essential  singularity,  the  value  of  the 
reciprocal  of  the  function  is  not  determinate  there  ;  and  then  the  reciprocal, 
as  well  as  the  function,  has  the  point  for  an  essential  singularity. 

33.  It  may  be  remarked  at  once  that  there  must  be  at  least  one 
infinite  value  among  the  values  which  a  function  can  assume  at  an  essential 
singularity.  For  if/  (z)  cannot  be  infinite  at  a,  then  the  limit  of  (z  —  a)f  (z) 
is  zero  when  z  =  a,  no  matter  what  the  non-infinite  values  of  f  (z)  may  be, 
that  is,  the  limit  is  a  determinate  zero.  The  function  (z  —  a)f(z)  is  regular 
in  the  vicinity  of  a :  hence  by  the  foregoing  test  for  an  ordinary  point, 
the  point  a  is  ordinary  and  the  value  of  the  uniform  function  f(z)  is 

*  Weierstrass,  Abh.  aus  der  Functionenlehre,  p.  2,  to  whom  the  name  is  due,  calls  it  ausser- 
wesentliche  singuldre  Stelle ;  the  term  non-essential  is  suggested  by  Mr  Cathcart,  Harnack,  p.  148. 
t  Weierstrass,  I.e.,  calls  it  wesentliche  singulare  Stelle. 


54  CONTINUATIONS   OF   A   FUNCTION  [33. 

determinate,  contrary  to  hypothesis.  Hence  the  function  must  have  at  least 
one  infinite  value  at  an  essential  singularity. 

Further,  a  uniform  function  must  be  capable  of  assuming  any  value  C  at 
an  essential  singularity.  For  an  essential  singularity  of  /  (z)  is  also  an 

essential  singularity  of  /  (z)  —  G  and  therefore  also  of  ..     \_n  •     The  last 

function  must  have  at  least  one  infinite  value  among  the  values  that  it  can 
assume  at  the  point ;  and,  for  this  infinite  value,  we  have  /  (z)  —  C  at  the 
point,  so  that/(f)  assumes  the  assigned  value  C  at  the  essential  singularity*. 

34.  Let  f(z)  denote  the  function  represented  by  a  series  of  powers 
Pj  (z  —  a),  the  circle  of  convergence  of  which  is  the  domain  of  the  ordinary 
point  a  of  the  function.  The  region  over  which  the  function  /  (z)  is  holo- 
morphic  may  extend  beyond  the  domain  of  a,  although  the  circumference 
bounding  that  domain  is  the  greatest  of  centre  a  that  can  be  drawn  within 
the  region.  The  region  evidently  cannot  extend  beyond  the  domain  of  a  in 
all  directions. 

Take  an  ordinary  point  b  in  the  domain  of  a.  The  value  at  b  of  the 
function /(V)  is  given  by  the  series  Pj  (b  —  a),  and  the  values  at  b  of  all  its 
derivatives  are  given  by  the  derived  series.  All  these  series  converge  within 
the  domain  of  a  and  they  are  therefore  finite  at  b ;  and  their  expressions 
involve  the  values  at  a  of  the  function  and  its  derivatives. 

Let  the  domain  of  b  be  formed.  The  domain  of  b  may  be  included  in 
that  of  a,  and  then  its  bounding  circle  will  touch  the  bounding  circle  of  the 
domain  of  a  internally.  If  the  domain  of  b  be  not  entirely  included  in  that 
of  a,  part  of  it  will  lie  outside  the  domain  of  a ;  but  it  cannot  include  the 
whole  of  the  domain  of  a  unless  its  bounding  circumference  touch  that  of  the 
domain  of  a  externally,  for  otherwise  it  would  extend  beyond  a  in  all 
directions,  a  result  inconsistent  with  the  construction  of  the  domain  of  a. 
Hence  there  must  be  points  excluded  from  the  domain  of  a  which  are  also 
excluded  from  the  domain  of  b. 

For  all  points  z  in  the  domain  of  b,  the  function  can  be  represented  by  a 
series,  say  P2  (2  —  b),  the  coefficients  of  which  are  the  values  at  b  of  the 
function  and  its  derivatives.  Since  these  values  are  partially  dependent 
upon  the  corresponding  values  at  a,  the  series  representing  the  function  may 
be  denoted  by  P2  (z  —  b,  a). 

At  a  point  z  in  the  domain  of  b  lying  also  in  the  domain  of  a,  the  two 
series  Pl  (z  —  a)  and  P2  (z  —  b,  a)  must  furnish  the  same  value  for  the 
function /  (V) ;  and  therefore  no  new  value  is  derived  from  the  new  series  P2 

*  Weierstrass,  I.e.,  pp.  50—52;  Durege,  Elemente  der  Theorie  der  Funktionen,  p.  119;  Holder, 
Math.  Ann.,  t.  xx,  (1882),  pp.  138 — 143  ;  Picard,  "  Memoire  sur  les  fonctions  entieres,"  Annahs  de 
VEcole  Norm.  Sup.,  2me  Ser.,  t.  ix,  (1880),  pp.  145 — 166,  which,  in  this  regard,  should  be  consulted 
in  connection  with  the  developments  in  Chapter  V.  See  also  §  62. 


34.]  OVER   ITS   REGION   OF   CONTINUITY  55 

which  cannot  be  derived  from  the  old  series  Pj.  For  all  such  points  the  new 
series  is  of  no  advantage ;  and  hence,  if  the  domain  of  b  be  included  in  that 
of  a,  the  construction  of  the  series  P2  (z  —  b,  a)  is  superfluous.  Hence  in 
choosing  the  ordinary  point  b  in  the  domain  of  a  we  choose  a  point,  if 
possible,  that  will  not  have  its  domain  included  in  that  of  a. 

At  a  point  z  in  the  domain  of  b,  which  does  not  lie  in  the  domain  of  a, 
the  series  P2  (z  —  b,  a)  gives  a  value  for  f(z)  which  cannot  be  given  by 
Pl  (z  —  a).  The  new  series  P2  then  gives  an  additional  representation  of  the 
function ;  it  is  called*  a  continuation  of  the  series  which  represents  the  function 
in  the  domain  of  a.  The  derivatives  of  P2  give  the  values  of  f(z)  for  points 
in  the  domain  of  b. 

It  thus  appears  that,  if  the  whole  of  the  domain  of  b  be  not  included  in 
that  of  a,  the  function  can,  by  the  series  which  is  valid  over  the  whole 
of  the  new  domain,  be  continued  into  that  part  of  the  new  domain  excluded 
from  the  domain  of  a. 

Now  take  a  point  c  within  the  region  occupied  by  the  combined  domains 
of  a  and  b ;  and  construct  the  domain  of  c.  In  the  new  domain,  the  function 
can  be  represented  by  a  new  series,  say  P3(z  —  c),  or,  since  the  coefficients 
(being  the  values  at  c  of  the  function  and  of  its  derivatives)  involve  the 
values  at  a  and  possibly  also  the  values  at  b  of  the  function  and  of  its 
derivatives,  the  series  representing  the  function  may  be  denoted  by 
Pz(z  —  c,  a,  b).  Unless  the  domain  of  c  include  points,  which  are  not 
included  in  the  combined  domains  of  a  and  b,  the  series  P3  does  not  give 
a  value  of  the  function  which  cannot  be  given  by  Pj  or  P2:  we  therefore 
choose  c,  if  possible,  so  that  its  domain  will  include  points  not  included  in 
the  earlier  domains.  At  such  points  z  in  the  domain  of  c  as  are  excluded 
from  the  combined  domains  of  a  and  6,  the  series  P3  (z  —  c,  a,  b)  gives  a  value 
for  f(z)  which  cannot  be  derived  from  P1  or  P2 ;  and  thus  the  new  series 
is  a  continuation  of  the  earlier  series. 

Proceeding  in  this  manner  by  taking  successive  points  and  constructing 
their  domains,  we  can  reach  all  parts  of  the  plane  connected  with  one 
another  where  the  function  preserves  its  holomorphic  character;  their 
combined  aggregate  is  called -f  the  region  of  continuity  of  the  function. 
With  each  domain,  constructed  so  as  to  include  some  portion  of  the  region  of 
continuity  not  included  in  the  earlier  domains,  a  series  is  associated,  which  is 
a  continuation  of  the  earlier  series  and,  as  such,  gives  a  value  of  the  function 
not  deducible  from  those  earlier  series ;  and  all  the  associated  series  are 
ultimately  derived  from  the  first. 

*  Biermann,   Theorie   der  analytischen  Functional,   p.   170,    which    may   be    consulted    in 
connection  with  the  whole  of  §  34;  the  German  word  is  Fortsetzung. 
t  Weierstrass,  I.e.,  p.  1. 


56  DEFINITION    OF   ANALYTIC    FUNCTION  [34. 

Each  of  the  continuations  is  called  an  Element  of  the  function.  The 
aggregate  of  all  the  distinct  elements  is  called  a  monogenic  analytic  function : 
it  is  evidently  the  complete  analytical  expression  of  the  function  in  its  region 
of  continuity. 

Let  z  be  any  point  in  the  region  of  continuity,  not  necessarily  in  the 
circle  of  convergence  of  the  initial  element  of  the  function;  a  value  of  the 
function  at  z  can  be  obtained  through  the  continuations  of  that  initial 
element.  In  the  formation  of  each  new  domain  (and  therefore  of  each  new 
element)  a  certain  amount  of  arbitrary  choice  is  possible ;  and  there  may, 
moreover,  be  different  sets  of  domains  which,  taken  together  in  a  set,  each 
lead  to  z  from  the  initial  point.  When  the  analytic  function  is  uniform,  as 
before  defined  (§  12),  the  same  value  at  z  for  the  function  is  obtained, 
whatever  be  the  set  of  domains.  If  there  be  two  sets  of  elements,  differently 
obtained,  which  give  at  z  different  values  for  the  function,  then  the  ana 
lytic  function  is  multiform,  as  before  defined  (§  12) ;  but  not  every  change 
in  a  set  of  elements  leads  to  a  change  in  the  value  at  z  of  a  multiform 
function,  and  the  analytic  function  is  uniform  within  such  a  region  of  the 
plane  as  admits  only  equivalent  changes  of  elements. 

The  whole  process  is  reversible  when  the  function  is  uniform.  We  can 
pass  back  from  any  point  to  any  earlier  point  by  the  use,  if  necessary,  of 
intermediate  points.  Thus,  if  the  point  a  in  the  foregoing  explanation 
be  not  included  in  the  domain  of  b  (there  supposed  to  contribute  a  continu 
ation  of  the  first  series),  an  intermediate  point  on  a  line,  drawn  in  the 
region  of  continuity  so  as  to  join  a  and  b,  would  be  taken ;  and  so  on, 
until  a  domain  is  formed  which  does  include  a.  The  continuation,  associated 
with  this  domain,  must  give  at  a  the  proper  value  for  the  function  and  its 
derivatives,  and  therefore  for  the  domain  of  a  the  original  series  Pl(z  —  a) 
will  be  obtained,  that  is,  Pj  (z  —  a)  can  be  deduced  from  P2  (z  —  b,  a)  the 
series  in  the  domain  of  b.  This  result  is  general,  so  that  any  one  of  the 
continuations  of  a  uniform  function,  represented  by  a  power-series,  can  be 
derived  from  any  other;  and  therefore  the  expression  of  such  a  function  in 
its  region  of  continuity  is  potentially  given  by  one  element,  for  all  the 
distinct  elements  can  be  derived  from  any  one  element. 

35.  It  has  been  assumed  that  the  property,  characteristic  of  some  of  the 
functions  adduced  as  examples,  of  possessing  either  accidental  or  essential 
singularities,  is  characteristic  of  all  functions ;  it  will  be  proved  (§  40)  to  hold 
for  every  uniform  function  which  is  not  a  mere  constant. 

The  singularities  limit  the  region  of  continuity ;  for  each  of  the  separate 
domains  is,  from  its  construction,  limited  by  the  nearest  singularity,  and  the 
combined  aggregate  of  the  domains  constitutes  the  region  of  continuity  when 


35.] 


SCHWARZ  S   CONTINUATION 


57 


they  form  a  continuous  space*.     Hence  the  complete  boundary  of  the  region 
of  continuity  is  the  aggregate  of  the  singularities  of  the  function-}-. 

It  may  happen  that  a  function  has  no  singularity  except  at  infinity ;  the 
region  of  continuity  then  extends  over  the  whole  finite  part  of  the  plane  but 
it  does  not  include  the  point  at  infinity. 

It  follows  from  the  foregoing  explanations  that,  in  order  to  know  a 
uniform  analytic  function,  it  is  necessary  to  know  some  element  of  the 
function,  which  has  been  shewn  to  be  potentially  sufficient  for  the  derivation 
of  the  full  expression  of  the  function  and  for  the  construction  of  its  region  of 
continuity. 

36.  The  method  of  continuation  of  a  function,  which  has  just  been 
described,  is  quite  general ;  there  is  one  particular  continuation,  which  is 
important  in  investigations  on  conformal  representations.  It  is  contained  in 
the  following  proposition,  due  to  SchwarzJ : — 

If  an  analytic  function  w  of  z  be  defined  only  for  a  region  8'  in  the 
positive  half  of  the  z-plane  and  if  continuous  real  values  of  w  correspond  to 
continuous  real  values  of  z,  then  w  can  be  continued  across  the  axis  of  real 
quantities. 

Consider  a  region  8",  symmetrical  with  S'  relative  to  the  axis  of  real 
quantities  (fig.  12).     Then  a  function  is 
defined  for  the  region  S"  by  associating 
a  value  w0,  the  conjugate  of  w,  with  z0, 
the  conjugate  of  z. 

Let  the  two  regions  be  combined  along 
the  portion  of  the  axis  of  ac  which  is  their 
common  boundary ;  they  then  form  a 
single  region  S'  +  S". 

Consider  the  integrals 


Fig.  12. 


1  [  w  j  A  !  [  wo 
o  —  •  I  —  i-dz  and  ^  —  -.  /  — 
fcp/fjr-f  2w»./,r«t- 


taken   round   the    boundaries   of  8'   and    of  8"   respectively.      Since   w    is 

*  Cases  occur  in  which  the  region  of  continuity  of  a  function  is  composed  of  isolated  spaces, 
each  continuous  in  itself,  but  not  continuous  into  one  another.  The  consideration  of  such  cases 
will  be  dealt  with  briefly  hereafter,  and  they  are  assumed  excluded  for  the  present  :  meanwhile, 
it  is  sufficient  to  note  that  each  continuous  space  could  be  derived  from  an  element  belonging  to 
some  domain  of  that  space  and  that  a  new  element  would  be  needed  for  a  new  space. 

t  See  Weierstrass,  I.e.,  pp.  1—3  ;  Mittag-Leffler,  "  Sur  la  representation  analytique  des  fonctions 
monogenes  uniformes  d'une  variable  independante,"  Acta  Math.,  t.  iv,  (1884),  pp.  1  et  seq., 
especially  pp.  1  —  8. 

£  Crelle,  t.  Ixx,  (1869),  pp.  106,  107,  and  Ges.  Math.  Abh.,  t.  ii,  pp.  66—68.  See  also  Darboux, 
Theorie  generate  des  surfaces,  t.  i,  §  130. 


58  SCHWARZ'S   CONTINUATION  [36. 

continuous  over  the  whole  area  of  8'  as  well  as  along  its  boundary  and 
likewise  w0  relative  to  8",  it  follows  that,  if  the  point  f  be  in  8',  the  value  of 
the  first  integral  is  w  (f  )  and  that  of  the  second  is  zero  ;  while,  if  £  lie  in  8", 
the  value  of  the  first  integral  is  zero  and  that  of  the  second  is  w0  (£).  Hence 
the  sum  of  the  two  integrals  represents  a  unique  function  of  a  point  in  either 
8'  or  8".  But  the  value  of  the  first  integral  is 


M  wdz      J^  [B  w  Q)  dap 

I  ("'  ~~  (f    •      C\        '    I  V> 

J  ji  2—  £      ZTriJ  A     x  —  L, 


the  first  being  taken  along  the  curve  EC.  A  and  the  second  along  the  axis 
AxB  ;  and  the  value  of  the  second  integral  is 

1    CAw0(x)dx       1_  f  *  W0dz0 
2-Tri  J  B    x  —  £         ZTTI  J   A  *o  —  £  ' 
the  first  being  taken  along  the  axis  Ex  A  and  the  second  along  the  curve 

ADB.     But 

w0  (ac)  =  w  (x), 

because  conjugate  values  w  and  w0  correspond  to  conjugate  values  of  the 
argument  by  definition  of  W0  and  because  w  (and  therefore  also  w0)  is  real 
and  continuous  when  the  argument  is  real  and  continuous.  Hence  when  the 
sum  of  the  four  integrals  is  taken,  the  two  integrals  corresponding  to  the 
two  descriptions  of  the  axis  of  x  cancel  and  we  have  as  the  sum 

wdz        1 


A 


and  this  sum  represents  a  unique  function  of  a  point  in  8'  +  8".  These  two 
integrals,  taken  together,  are 

_L  [w'dz 
2Tn]z-t' 

taken  round  the  whole  contour  of  8'  +  8",  where  w'  is  equal  to  w  (f)  in  the 
positive  half  of  the  plane  and  to  w0  (^)  in  the  negative  half. 

For  all  points  £  in  the  whole  region  8'  +  8",  this  integral  represents  a 
single  uniform,  finite,  continuous  function  of  f;  its  value  is  w  (£)  in  the 
positive  half  of  the  plane  and  is  w0  (f)  in  the  negative  half;  and  therefore 
w0  (£)  is  the  continuation  into  the  negative  half  of  the  plane  of  the  function, 
which  is  defined  by  w  (£)  for  the  positive  half. 

For  a  point  c  on  the  axis  of  x,  we  have 

w  (z)  -w(c)  =  A(z-c)  +  B(z-cy>+C(z-cY  +  ...; 

and  all  the  coefficients  A,  B,  C,...  are  real.  If,  in  addition,  w  be  such  a 
function  of  z  that  the  inverse  functional  relation  makes  z  a  uniform 
analytic  function  of  w,  it  is  easy  to  see  that  A  must  not  vanish,  so  that  the 
functional  relation  may  be  expressed  in  the  form 

w(z)—w  (c)  =  (z-c}P(z-  c), 
where  P  (z  —  c)  does  not  vanish  when  z  =  c. 


CHAPTER    IV. 

GENERAL  PROPERTIES  OF  UNIFORM  FUNCTIONS,  PARTICULARLY  OF  THOSE 
WITHOUT  ESSENTIAL  SINGULARITIES. 

37.  IN  the  derivation  of  the  general  properties  of  functions,  which  will  be 
deduced  in  the  present  and  the  next  three  chapters  from  the  results  already 
obtained,  it  is  to  be  supposed,  in  the  absence  of  any  express  statement  to 
other  effect,  that  the  functions  are  uniform,  monogenic  and,  except  at  either 
accidental  or  essential  singularities,  continuous*. 

THEOREM  I.  A  function,  which  is  constant  throughout  any  region  of  the 
plane  not  infinitesimal  in  area,  or  which  is  constant  along  any  line  not  infini 
tesimal  in  length,  is  constant  throughout  its  region  of  continuity. 

For  the  first  part  of  the  theorem,  we  take  any  point  a  in  the  region  of  the 
plane  where  the  function  is  constant,  and  we  draw  a  circle  of  centre  a  and 
of  any  radius,  provided  only  that  the  circle  remains  within  the  region  of 
continuity  of  the  function.  At  any  point  z  within  this  circle  we  have 

/<*)  =/(a)  +  (z  -  a)f  (a)  +  (-, ~^ f"  (a)  +  . . , 

a  converging  series  the  coefficients  of  which  are  the  values  of  the  function 
and  its  derivatives  at  a.  But 

/X«)  =  Limit  of  ^±MZ/^),     :.        V,  '         :• 

which  is  zero  because  f(a  +  Ba)  is  the  same  constant  as  f(a)  :  so  that  the 
first  derivative  is  zero  at  a.  Similarly,  all  the  derivatives  can  be  shewn  to 
be  zero  at  a ;  hence  the  above  series  after  its  first  term  is  evanescent, 
and  we  have 

/(*)-/<«), 

that  is,  the  function  preserves  its  constant  value  throughout  its  region  of 
continuity. 

The  second  result  follows  in  the  same  way,  -when  once  the  derivatives  are 
proved  zero.  Since  the  function  is  monogenic,  the  value  of  the  first  and 

*  It  will  be  assumed,  as  in  §  35  (note,  p.  57),  that  the  region  of  continuity  consists  of  a  single 
space ;  functions,  with  regions  of  continuity  consisting  of  a  number  of  separated  spaces,  will  be 
discussed  in  Chap.  VII. 


60  ZEROS    OF    A  [37. 

of  each  of  the  successive  derivatives  will  be  obtained,  if  we  make  the 
differential  element  of  the  independent  variable  vanish  along  the  line. 

Now,  if  a  be  a  point  on  the  line  and  a  +  8a  a  consecutive  point,  we  have 
f(a  +  So)  =  f(a)  ;  hence  /'  (a)  is  zero.  Similarly  the  first  derivative  at  any 
other  point  on  the  line  is  zero.  Therefore  we  have  /'  (a  +  So)  =f  (a),  for 
each  has  just  been  proved  to  be  zero :  hence  /"  (a)  is  zero  ;  and  similarly  the 
value  of  the  second  derivative  at  any  other  point  on  the  line  is  zero.  So  on 
for  all  the  derivatives :  the  value  of  each  of  them  at  a  is  zero. 

Using  the  same  expansion  as  before  and  inserting  again  the  zero  values 
of  all  the  derivatives  at  a,  we  find  that 

/(*)=/(«), 

so  that  under  the  assigned  condition  the  function  preserves  its  constant  value 
throughout  its  region  of  continuity. 

It  should  be  noted  that,  if  in  the  first  case  the  area  be  so  infinitesimally 
small  and  in  the  second  the  line  be  so  infinitesimally  short  that  consecutive 
points  cannot  be  taken,  then  the  values  at  a  of  the  derivatives  cannot  be 
proved  to  be  zero  and  the  theorem  cannot  then  be  inferred. 

COROLLARY  I.  If  two  functions  have  the  same  value  over  any  area  of 
their  common  region  of  continuity  which  is  not  infinitesimally  small  or  along 
any  line  in  that  region  which  is  not  infinitesimally  short,  then  they  have  the 
same  values  at  all  points  in  their  common  region  of  continuity. 

This  is  at  once  evident :  for  their  difference  is  zero  over  that  area  or  along 
that  line  and  therefore,  by  the  preceding  theorem,  their  difference  has  a 
constant  zero  value,  that  is,  the  functions  have  the  same  values,  everywhere 
in  their  common  region  of  continuity. 

But  two  functions  can  have  the  same  values  at  a  succession  of  isolated 
points,  without  having  the  same  values  everywhere  in  their  common  region 
of  continuity ;  in  such  a  case  the  theorem  does  not  apply,  the  reason  being 
that  the  fundamental  condition  of  equality  over  a  continuous  area  or  along 
a  continuous  line  is  not  satisfied. 

COROLLARY  II.  A  function  cannot  be  zero  over  any  continuous  area  of  its 
region  of  continuity  which  is  not  infinitesimal  or  along  any  line  in  that  region 
which  is  not  infinitesimally  short  without  being  zero  everywhere  in  its  region  of 
continuity. 

This  corollary  is  deduced  in  the  same  manner  as  that  which  precedes. 

If,  then,  there  be  a  function  which  is  evidently  not  zero  everywhere,  we 
conclude  that  its  zeros  are  isolated  points  though  such  points  may  be  multiple 
zeros. 

Further,  in  any  finite  area  of  the  region  of  continuity  of  a  function  that  is 
subject  to  variation,  there  can  be  at  most  only  a  finite  number  of  its  zeros,  when 


37.]  UNIFORM   FUNCTION  61 

no  point  of  the  boundary  of  the  area  is  infinitesimally  near  an  essential 
singularity.  For  if  there  were  an  infinite  number  of  such  points  in  any 
such  region,  there  must  be  a  cluster  in  at  least  one  area  or  a  succession 
along  at  least  one  line,  infinite  in  number  and  so  close  as  to  constitute  a 
continuous  area  or  a  continuous  line  where  the  function  is  everywhere  zero. 
This  would  require  that  the  function  should  be  zero  everywhere  in  its  region 
of  continuity,  a  condition  excluded  by  the  hypothesis. 

And  it  immediately  follows  that  the  points  (other  than  those  infini 
tesimally  near  an  essential  singularity)  in  a  region  of  continuity,  at  which  a 
function  assumes  any  the  same  value,  are  isolated  points  ;  and  that  only  a 
finite  number  of  such  points  occur  in  any  finite  area. 

38.  THEOREM  II.  The  multiplicity  m  of  any  zero  a  of  a  function  is 
finite  provided  the  zero  be  an  ordinary  point  of  the  function,  which  is  not  zero 
throughout  its  region  of  continuity;  and  the  function  can  be  expressed  in  the 


where  <f>  (z)  is  holomorphic  in  the  vicinity  of  a,  and  a  is  not  a  zero  of  <£  (z). 

Let  f(z)   denote   the   function  ;   since  a   is   a   zero,   we    have  f(a)  =  0. 
Suppose  that  /'(a),  f"  (a),  ......  vanish:  in  the  succession  of  the  derivatives 

of  f,  one  of  finite  order  must  be  reached  which  does  not  have  a  zero  value. 
Otherwise,  if  all  vanish,  then  the  function  and  all  its  derivatives  vanish  at  a; 
the  expansion  of  f(z)  in  powers  of  z  —  a  leads  to  zero  as  the  value  of  f  (z\ 
that  is,  the  function  is  everywhere  zero  in  the  region  of  continuity,  if  all  the 
derivatives  vanish  at  a. 

Let,  then,  the  wth  derivative  be  the  first  in  the  natural  succession  which 
does  not  vanish  at  a,  so  that  m  is  finite.  Using  Cauchy's  expansion,  we  have 

(?  —  n\tm)  (  ~  _  n\(m+\) 

f(z)  =  (Z     a      /«  (a)  +  S£Za_/F*  (a)  +  .  .  . 

J  m  !      J  (m  +  1)  !  J 

=  (z-ay*$(z\ 

where  <£  (z)  is  a  function  that  does  not  vanish  with  a  and,  being  the  quotient 
of  a  converging  series  by  a  monomial  factor,  is  holomorphic  in  the  immediate 
vicinity  of  a. 

COROLLARY  I.  If  infinity  be  a  zero  of  a  function  of  multiplicity  m  and 
at  the  same  time  be  an  ordinary  point  of  the  function,  then  the  function  can  be 


expressed  in  the  form  z~m  $  f-J , 


where  </>(-)  is  a  function  that  is  continuous  and  non-evanescent  for  infinitely 


large  values  of  z. 

The  result  can  be  derived  from  the  expansion  in  §  30  in  the  same  way  as 
the  foregoing  theorem  from  Cauchy's  expansion. 


62  ZEROS   OF   A  [38. 

COROLLARY  II.  The  number  of  zeros  of  a  function,  account  being  taken  of 
their  multiplicity,  which  occur  within  a  finite  area  of  the  region  of  continuity 
of  the  function,  is  finite,  when  no  point  of  the  boundary  of  the  area  is  infinitesi- 
mally  near  an  essential  singularity. 

By  Corollary  II.  of  §  37,  the  number  of  distinct  zeros  in  the  limited  area 
is  finite,  and,  by  the  foregoing  theorem,  the  multiplicity  of  each  is  finite ; 
hence,  when  account  is  taken  of  their  respective  multiplicities,  the  total 
number  of  zeros  is  still  finite. 

The  result  is,  of  course,  a  known  result  for  an  algebraical  polynomial ;  but 
the  functions  in  the  enunciation  are  not  restricted  to  be  of  the  type  of 
algebraical  polynomials. 

Note.  It  is  important  to  notice,  both  for  the  Theorem  and  for  Corollary  I, 
that  the  zero  is  an  ordinary  point  of  the  function  under  consideration ;  the 
implication  therefore  is  that  the  zero  is  a  definite  zero  and  that  in  the 
immediate  vicinity  of  the  point  the  function  can  be  represented  in  the  form 

P(z  —  a)  or  P  [-] ,  the  function  P(a  —  a)  or  P  (— )  being  .always  a  definite 

\<6  /  \        / 

zero. 

Instances  do  occur  for  which  this  condition  is  not  satisfied.  The  point 
may  not  be  an  ordinary  point,  and  the  zero  value  may  be  an  indeterminate 
zero ;  or  zero  may  be  only  one  of  a  set  of  distinct  values  though  everywhere 
in  the  vicinity  the  function  is  regular.  Thus  the  analysis  of  §  13  shews  that 

z=a  is  a  point  where  the  function  sn  -    -  has  any  number  of  zero  values  and 

Z       CL 

any  number  of  infinite  values,  and  there  is  no  indication  that  there  are  not 
also  other  values  at  the  point.  In  such  a  case  the  preceding  proposition  does 
not  apply  ;  there  may  be  no  limit  to  the  order  of  multiplicity  of  the  zero,  and 
we  certainly  cannot  infer  that  any  finite  integer  m  can  be  obtained  such  that 

(z  -  a)~m  <j>  (z) 

is  finite  at  the  point.  Such  a  point  is  (§  32)  an  essential  singularity  of  the 
function. 

39.  THEOREM  III.  A  multiple  zero  of  a  function  is  a  zero  of  its 
derivative ;  and  the  multiplicity  for  the  derivative  is  less  or  is  greater  by 
unity  according  as  the  zero  is  not  or  is  at  infinity. 

If  a  be  a  point  in  the  finite  part  of  the  plane  which  is  a  zero  of  f(z) 
of  multiplicity  n,  we  have 

/(f)-(*T.a)»  +  («X 

and  therefore         /'  (z)  =  (z  -  a)n~l  [n$  (z}  +  (z-a)  $  (z)}. 
The  coefficient  of  (z  —  a)n~l  is  holomorphic  in  the  immediate  vicinity  of  a  and 
does  not  vanish  for  a ;  hence  a  is  a  zero  for  /'  (z)  of  decreased  multiplicity 


39.]  UNIFORM   FUNCTION 

If  z  =  oo  be  a  zero  off(z)  of  multiplicity  r,  then 


where  <£  (-)  is  holomorphic  for  very  large  values  of  z  and  does  not  vanish  at 

\z  / 

infinity.     Therefore 


The  coefficient  of  ^~r~1  is  holomorphic  for  very  large  values  of  z,  and  does 
not  vanish  at  infinity  ;  hence  z=<x>  is  a  zero  off  (z)  of  increased  multiplicity 
r  +  l. 

Corollary  I.  If  a  function  be  finite  at  infinity,  then  z  =  oo  is  a  zero  of  the 
first  derivative  of  multiplicity  at  least  two. 

Corollary  II.     If  a  be  a  finite  zero  off(z)  of  multiplicity  n,  we  have 
f(z)=     n        #(z) 
f(z)      ir-**  fW 

Now  a  is  not  a  zero  of  <J>  (z)  ;  and  therefore  ^4^r  is  finite,  continuous,  uniform 

9W 

and  monogenic  in  the  immediate  vicinity  of  a.  Hence,  taking  the  integral 
of  both  members  of  the  equation  round  a  circle  of  centre  a  and  of  radius 
so  small  as  to  include  no  infinity  and  no  zero,  other  than  a,  of  /  (z)  _  and 
therefore  no  zero  of  $(z)  —  we  have,  by  §  17  and  Ex.  2,  §  25, 

~jT/   \  ^"^  ~  ^- 

/(*) 


40.     THEOREM  IV.     A  function  must  have  an  infinite  value  for  some  finite 
or  infinite  value  of  the  variable. 

If  M  be  a  finite  maximum  value  of  the  modulus  for  points  in  the  plane, 
then  (§  22)  we  have 


where  r  is  the  radius  of  an  arbitrary  circle  of  centre  a,  provided  the  whole  of 
the  circle  is  in  the  region  of  continuity  of  the  function.  But  as  the  function 
is  uniform,  monogenic,  finite  and  continuous  everywhere,  this  radius  can  be 
increased  indefinitely ;  when  this  increase  takes  place,  the  limit  of 

is  zero  and  therefore  /<»>  (a)  vanishes.  This  is  true  for  all  the  indices  1,2,... 
of  the  derivatives. 


64  INFINITIES   OF   A  [40. 

Now  the  function  can  be  represented  at  any  point  z  in  the  vicinity  of  a 
by  the  series 


which  degenerates,  under  the  present  hypothesis,  to  /(a),  so  that  the  function 
is  everywhere  constant.  Hence,  if  a  function  has  not  an  infinity  somewhere 
in  the  plane,  it  must  be  a  constant. 

The  given  function  is  not  a  constant;  and  therefore  there  is  no  finite 
limit  to  the  maximum  value  of  its  modulus,  that  is,  the  function  acquires 
an  infinite  value  somewhere  in  the  plane. 

COROLLARY  I.  A  function  must  have  a  zero  value  for  some  finite  or 
infinite  value  of  the  variable. 

For  the  reciprocal  of  a  uniform  monogenic  analytic  function  is  itself  a 
uniform  monogenic  analytic  function  ;  and  the  foregoing  proposition  shews 
that  this  reciprocal  must  have  an  infinite  value  for  some  value  of  the 
variable,  which  therefore  is  a  zero  of  the  function. 

COROLLARY  II.     A  function  must  assume  any  assigned  value  at  least  once. 

COROLLARY  III.  Every  function  which  is  not  a  mere  constant  must  have 
at  least  one  singularity,  either  accidental  or  essential.  For  it  must  have 
an  infinite  value  :  if  this  be  a  determinate  infinity,  the  point  is  an  accidental 
singularity  (§  32)  ;  if  it  be  an  infinity  among  a  set  of  values  at  the  point,  the 
point  is  an  essential  singularity  (§§  32,  33). 

41.  Among  the  infinities  of  a  function,  the  simplest  class  is  that  con 
stituted  by  its  accidental  singularities,  already  defined  (§  32)  by  the  property 
that,  in  the  immediate  vicinity  of  such  a  point,  the  reciprocal  of  the  function 
is  regular,  the  point  being  an  ordinary  (zero)  point  for  that  reciprocal. 

THEOREM  V.  A  function,  which  has  a  point  cfor  an  accidental  singularity, 
can  be  expressed  in  the  foi*m 

(z  -  c}~n  (f>  (z), 

where  n  is  a  finite  positive  integer  and  <f>  (z)  is  a  continuous  function  in  the 
vicinity  of  c. 

Since  c  is  an  accidental  singularity  of  the  function  f(z},  the  function  ^y-r 

/  (z) 

is  regular  in  the  vicinity  of  c  and  is  zero  there  (§  32).  Hence,  by  §  38,  there 
is  a  finite  limit  to  the  multiplicity  of  the  zero,  say  n  (which  is  a  positive 
integer),  and  we  have 


where  ^  (z)  is  uniform,  monogenic  and  continuous  in  the  vicinity  of  c  and  is 
not  zero  there.     The  reciprocal  of  ^  (z),  say  <f>  (z),  is  also  uniform,  monogenic 


41.]  UNIFORM   FUNCTION  65 

and  continuous  in  the  vicinity  of  c,  which  is  an  ordinary  point  for  (f>  (z)  ; 
hence  we  have 

f(z}  =  (Z-c)-^(z\ 

which  proves  the  theorem. 

The  finite  positive  integer  n  measures  the  multiplicity  of  the  accidental 
singularity  at  c,  which  is  sometimes  said  to  be  of  multiplicity  n  or  of 
order  n. 

Another  analytical  expression  for  f(z)  can  be  derived  from  that  which 
has  just  been  obtained.  Since  c  is  an  ordinary  point  for  <f>  (z)  and  not  a  zero, 
this  function  can  be  expanded  in  a  series  of  ascending,  positive,  integral 
powers  of  z  —  c,  converging  in  the  vicinity  of  c,  in  the  form 

£(*)  =  P(*-c) 

=  uQ  +  ul(z-c}  +  ...  +  un^(z-c)n-l+un(z-c)n+... 

=  u0  +  u,(z  -  c)  +  ...  +  un_^(z  -  c)71-1  +  (z-  c)nQ(z-c), 
where  Q(z  —  c),  a  series  of  positive,  integral,  powers  of  z  —  c  converging  in  the 
vicinity  of  c,  is  a  monogenic  analytic  function  of  z.     Hence  we  have 

^  =  ^»  +  (7^+  -  +,~;  +  «('-')> 

the  indicated  expression  for  f(z),  valid  in  the  immediate  vicinity  of  c,  where 
Q  (z  —  c)  is  uniform,  finite,  continuous  and  monogenic. 

COROLLARY.  A  function,  which  has  z=  oo  for  an  accidental  singularity  of 
multiplicity  n,  can  be  expressed  in  the  form 


_ 

where  </>(-)  is  a  continuous  function  for  very  large  values  of  \z  ,  and  is  not 

\zj 

zero  when  z  =  oo  .     It  can  also  be  expressed  in  the  form 

1  +  ...  +  an^  z  +  Q  (-}  , 
\zj 


where  Q  (  -  j  is  uniform,  finite,  continuous  and  monogenic  for  very  large  values 

f\»\. 

The  derivation  of  the  form  of  the  function  in  the  vicinity  of  an  accidental 
singularity  has  been  made  to  depend  upon  the  form  of  the  reciprocal  of  the 
function.  Whatever  be  the  (finite)  order  of  that  point  as  a  zero  of  the 
reciprocal,  it  is  assumed  that  other  zeros  of  the  reciprocal  are  not  at  merely 
infinitesimal  distances  from  the  point,  that  is,  that  other  infinities  of  the 
function  are  not  at  merely  infinitesimal  distances  from  the  point. 

Hence  the  accidental  singularities  of  a  function  are  isolated  points  ;  and 
there  is  only  a  finite  number  of  them  in  any  limited  portion  of  the  plane. 
F.  5 


66  INFINITIES   OF   A  [42. 

42.  We  can  deduce  a  criterion  which  determines  whether  a  given  singu 
larity  of  a  function /(f)  is  accidental  or  essential. 

When  the  point  is  in  the  finite  part  of  the  plane,  say  at  c,  and  a  finite 
positive  integer  n  can  be  found  such  that 

is  not  infinite  at  c,  then  c  is  an  accidental  singularity. 

When  the  point  is  at  infinity  and  a  finite  positive  integer  n  can  be  found 
such  that 

is  not  infinite  when  z  =  oc  ,  then  z  =  oo  is  an  accidental  singularity. 

If  one  of  these  conditions  be  not  satisfied,  the  singularity  at  the  point  is 
essential.  But  it  must  not  be  assumed  that  the  failure  of  the  limitation  to 
finiteness  in  the  multiplicity  of  the  accidental  singularity  is  the  only  source 
or  the  complete  cause  of  essential  singularity. 

Since  the  association  of  a  single  factor  with  the  function  is  effective  in 
preventing  an  infinite  value  at  the  point  when  one  of  the  conditions  is 
satisfied,  it  is  justifiable  to  regard  the  discontinuity  of  the  function  at 
the  point  as  not  essential  and  to  call  the  singularity  either  non-essential 
or  accidental  (§  82). 

43.  THEOREM  VI.     The  poles  of  a  function,  that  lie  in  the  finite  part 
of  the  plane,  are  all  the  poles  (of  increased  multiplicity)  of  the  derivatives  of 
the  function  that  lie  in  the  finite  part  of  the  plane. 

Let  c  be  a  pole  of  the  function  f(z)  of  multiplicity  p :  then,  for  any  point 
z  in  the  vicinity  of  c, 

where  </>  (z)  is  holomorphic  in  the  vicinity  of  c,  and  does  not  vanish  for  z  =  c. 
Then  we  have 

f'(2)  =  (z~  c)~p  $'  (z)  ~  P  (2  ~  c)  p  1  $  W 
=  (z-c)-P-*{(z-c)<j>'(z)-p<}>(z)} 

where  %  (z)  is  holomorphic  in  the  vicinity  of  c,  and  does  not  vanish  for  z  =  c. 

Hence  c  is  a  pole  of/'  (z)  of  multiplicity  ^9  +  1.  Similarly  it  can  be  shewn 
to  be  a  pole  of /(r)  (z)  of  multiplicity  p  +  r. 

This  proves  that  all  the  poles  of  f(z)  in  the  finite  part  of  the  plane  are 
poles  of  its  derivatives.  It  remains  to  prove  that  a  derivative  cannot  have 
a  pole  which  the  original  function  does  not  also  possess. 

Let  a  be  a  pole  off'(z)  of  multiplicity  m :  then,  in  the  vicinity  of  a,f'(z) 
can  be  expressed  in  the  form 


43.]  UNIFORM   FUNCTION  £7 

where  ^  (z)  is  holomorphic  in  the  vicinity  of  a  and  does  not  vanish  for  z  =  a 
Thus 


and  therefore  f  (*)  =  -         .  +   j_  , 

y   v  '     JlV^    •<*-«)*"* 

so  that,  integrating,  we  have 

f(z}=      *(«)         _*>) 

m  0  -  a)™-1     (m  -  1)  0  -  a)™-2 
that  is,  a  is  a  pole  of/0). 

An  apparent  exception  occurs  in  the  case  when  m  is  unity:  for  then 
we  have 


the  integral  of  which  leads  to 

f(z}  =  ^  (a)  log  (z  -  a)  +  .  .  .  , 

so  that/0)  is  no  longer  uniform,  contrary  to  hypothesis.  Hence  a  derivative 
cannot  have  a  simple  pole  in  the  finite  part  of  the  plane  ;  and  so  the  exception 
is  excluded. 

The  theorem  is  thus  proved. 

COROLLARY  I.     The  rth  derivative  of  a  function  cannot  have  a  pole  in  the 
finite  part  of  the  plane  of  multiplicity  less  than  r  +  1. 

COROLLARY  II.     If  c  be  a  pole  of  f  (z)  of  any  order  of  multiplicity  ^  and 
if  f(r]  (z)  be  expressed  in  the  form 

,  _  Oi__ 

»       /  _.  _\.,  _!_*•  _  1        I       ••••••* 


(Z  -  CY+T       (Z- 

there  are  no  terms  in  this  expression  with  the  indices  -  1,  -  2,  ......  ,  -  r. 

COROLLARY  III.     If  c  be  a  pole  of/  (z)  of  multiplicity  p,  we  have 


= 
f(z)      z-c~*  4>(z)' 

where  $  (z)  is  a  holomorphic  function  that  does  not  vanish  for  z  =  c,  so  that 

<£'  0)  • 

-T-/JN  is  a  holomorphic  function  in  the  vicinity  of  c.     Taking  the  integral  of 

f'(z) 

-j-j~\  round  a  circle,  with  c  for  centre,  with  radius  so  small  as  to  exclude  all 

other  poles  or  zeros  of  the  function  f  (z),  we  have 


5—2 


(}8  INFINITIES    OF    A  [43. 

COROLLARY  IV.  If  a  simple  closed  curve  include  a  number  N  of  zeros  of 
a  uniform  function  f  (z)  and  a  number  P  of  its  poles,  in  both  of  which 
numbers  account  is  taken  of  possible  multiplicity,  and  if  the  curve  contain 
no  essential  singularity  of  the  function,  then 


the  integral  being  taken  round  the  curve. 

f  (z) 
The  only  infinities  of  the  function  '  ^i  within  the  curve  are  the  zeros 

j(z) 

and  the  poles  of  /  (z).  Round  each  of  these  draw  a  circle  of  radius  so  small 
as  to  include  it  but  no  other  infinity  ;  then,  by  Cor.  II.  §  18,  the  integral 
round  the  closed  curve  is  the  sum  of  the  values  when  taken  round  these 
circles.  By  the  Corollary  II.  §  39  and  by  the  preceding  Corollary  III.,  the 
sum  of  these  values  is 

=  2w  —  %> 

=  N-P. 

It  is  easy  to  infer  the  known  theorem  that  the  number  of  roots  of  an 
algebraical  polynomial  of  order  n  is  n,  as  well  as  the  further  result  that 
2^  (N  -  P)  is  the  variation  of  the  argument  of  /  (z)  as  z  describes  the 
closed  curve  in  a  positive  sense. 

Ex.  Prove  that,  if  F(z)  be  holomorphic  over  an  area,  of  simple  contour,  which  con 
tains  roots  «!,  «2,...  of  multiplicity  m»  m2,...  and  poles  cx,  c2)...  of  multiplicity  p^  p2J... 
respectively  of  a  function  f(z)  which  has  no  other  singularities  within  the  contour,  then 


the  integral  being  taken  round  the  contour. 

In  particular,  if  the  contour  contains  a  single  simple  root  a  and  no  singularity,  then  that 
root  is  given  by 


the  integral  being  taken  as  before.     (Laurent.) 

44.  THEOREM  VII.  If  infinity  be  a  pole  of  f  (z),  it  is  also  a  pole  of 
f  (z)  only  when  it  is  a  multiple  pole  of  f  (z). 

Let  the  multiplicity  of  the  pole  for  f  (z)  be  ?i;  then  for  very  large  values 
of  z  we  have 

/(*)-*•*£), 

where  <j>  is  holomorphic  for  very  large  values  of  z  and  does  not  vanish  at 
infinity  ;   hence 

A«)**"  •*-*'• 


44.]  UNIFORM   FUNCTION  69 

The  coefficient  of  zn~*  is  holomorphic  for  very  large  values  of  z  and  does  not 
vanish  at  infinity  ;  hence  infinity  is  a  pole  of/'  (z}  of  multiplicity  n  —  1. 

If  n  be  unity,  so  that  infinity  is  a  simple  pole  of  /  (z),  then  it  is  not  a 
pole  of/'  (2);   the  derivative  is  then  finite  at  infinity. 

45.     THEOREM  VIII.     A  function,  which  has  no  singularity  in  a  finite 
part  of  the  plane,  and  has  z  =  oo  for  a  pole,  is  an  algebraical  polynomial. 

Let  n,  necessarily  a  finite  integer,  be  the  order  of  multiplicity  of  the  pole 
at  infinity  :  then  the  function  /  (z)  can  be  expressed  in  the  form 


1  +  ......  +an^z  +  Q  -   , 

\zJ 

where  Q  (-  J  is  a  holomorphic  function  for  very  large  values  of  z,  and  is  finite 
(or  zero)  when  z  is  infinite. 

Now  the  first  n  terms  of  the  series  constitute  a  function  which  has  no 
singularities  in  the  finite  part  of  the  plane  :  and  /  (z)  has  no  singularities 

in  that  part  of  the  plane.     Hence  Q  (  -  J  has  no  singularities  in  the  finite  part 

of  the  plane  :  it  is  finite  for  infinite  values  of  z.     It  thus  can  never  have  an 
infinite  value:  and  it  is  therefore  merely  a  constant,  say  an.     Then 

/  (z)  =  a,zn  +  a^-1  +  ......  +  an^z  +  an, 


a  polynomial  of  degree  equal   to   the  multiplicity  of  the  pole  at  infinity, 
supposed  to  be  the  only  pole  of  the  function. 

46.     The  above  result  may  be  obtained  in  the  following  manner. 

Since  z  =  GO  is  a  pole  of  multiplicity  n,  the  limit  of  z~nf  (z}  is  not  infinite 
when  z  =  oo  . 

Now  in  any  finite  part  of  the  plane  the  function  is  everywhere  finite,  so 
that  we  can  use  the  expansion 


where  £  =  *'"">     dt 


''+l    t-z' 

the  integral  being  taken  round  a  circle  of  any  radius  r  enclosing  the  point  z 
and  having  its  centre  at  the  origin.  As  the  subject  of  integration  is  finite 
everywhere  along  the  circumference,  we  have,  by  Darboux's  expression  in 
(IV.)  S  14, 


T»i  T  _  z 

where  r  is  some  point  on  the  circumference  and  X  is  a  quantity  of  modulus 
not  greater  than  unity. 


70  TRANSCENDENTAL   AND  [46. 

Let  T  =  reia- ;  then 


X  .  fM 

"•      71-4-1      °flii  »/      \      / 

'?*  rn 


r 


f(T\ 
By  definition,  the  limit  of      n     as  T  (and    therefore  r)  becomes   infinitely 

(£       -\—1 
1  --  e~ai  }     is  unity. 
r        J 

Since  \  is  not  greater  than  unity,  the  limit  of  \jr  in  the  same  case  is  zero  ; 
hence  with  indefinite  increase  of  r,  the  limit  of  R  is  zero  and  so 


shewing  as  before  that/(^)  is  an  algebraical  polynomial. 

47.  As  the  quantity  n  is  necessarily  a  positive  integer*,  there  are  two 
distinct  classes  of  functions  discriminated  by  the  magnitude  of  n. 

The  first  (and  the  simpler)  is  that  for  which  n  has  a  finite  value.  The 
polynomial  then  contains  only  a  finite  number  of  terms,  each  with  a  positive 
integral  index  ;  and  the  function  is  then  a  rational,  integral,  algebraical 
polynomial  of  degree  n. 

The  second  (and  the  more  extensive,  as  significant  functions)  is  that 
for  which  n  has  an  infinite  value.  The  point  z  =  oo  is  not  a  pole,  for  then 
the  function  does  not  satisfy  the  test  of  §  42  :  it  is  an  essential  singularity 
of  the  function,  which  is  expansible  in  an  infinite  converging  series 
of  positive  integral  powers.  To  functions  of  this  class  the  general  term 
transcendental  is  applied. 

The  number  of  zeros  of  a  function  of  the  former  class  is  known  :  it  is 
equal  to  the  degree  of  the  function.  It  has  been  proved  that  the  zeros  of  a 
transcendental  function  are  isolated  points,  occurring  necessarily  in  finite 
number  in  any  finite  part  of  the  region  of  continuity  of  the  function,  no 
point  on  the  boundary  of  the  part  being  infinitesimally  near  an  essential 
singularity  ;  but  no  test  has  been  assigned  for  the  determination  of  the  total 
number  of  zeros  of  a  function  in  an  infinite  part  of  the  region  of  con 
tinuity. 

Again,  when  the  zeros  of  a  polynomial  are  given,  a  product-expression  can 
at  once  be  obtained  that  will  represent  its  analytical  value.  Also  we  know 
that,  if  a  be  a  zero  of  any  uniform  analytic  function  of  multiplicity  n,  the 
function  can  be  represented  in  the  vicinity  of  a  by  the  expression 

(x-a}n<t>(z\ 

where  <£  (z)  is  holomorphic  in  the  vicinity  of  a.     The  other  zeros  of  the 
function  are  zeros  of  <f>  (z)  ;   this  process  of  modification  in  the  expression 

*  It  is  unnecessary  to  consider  the  zero  value  of  n,  for  the  function  is  then  a  polynomial  of 
order  zero,  that  is,  it  is  a  constant. 


47.]  ALGEBRAICAL   UNIFORM   FUNCTIONS  71 

can  be  continued  for  successive  zeros  so  long  as  the  number  of  zeros  taken 
account  of  is  limited.  But  when  the  number  of  zeros  is  unlimited,  then  the 
inferred  product-expression  for  the  original  function  is  not  necessarily  a 
converging  product;  and  thus  the  question  of  the  formal  factorisation  of  a 
transcendental  function  arises. 

48.     THEOREM  IX.     A  function,  all  the  singularities  of  which  are  accid 
ental,  is  a  rational,  algebraical,  meromorphic  function. 

Since    all    the    singularities    are   accidental,    each    must   be    of    finite 
multiplicity  ;  and  therefore  infinity,  if  an  accidental  singularity,  is  of  finite 
multiplicity.     All  the  other  poles  are  in  the  finite  part  of  the  plane  ;  they 
are  isolated  points  and  therefore  only  finite  in  number,  so  that  the  total 
number  of  distinct  poles  is  finite  and  each  is  of  finite  order.     Let  them  be 
«!,  a2,  ......  ,  a^  of  orders  m1}  m2,  ......  ,  m^  respectively  :  let  m  be  the  order  of 

the  pole  at  infinity:  and  let  the  poles  be  arranged  in  the  sequence  of 
decreasing  moduli  such  that  [aj  >  aF_!  >  ......  >|&i|- 

Then,  since  infinity  is  a  pole  of  order  m,  we  have 

/  0)  =  amzm  +  a^z™-1  +  ......  +  a^z  +  /„  <», 

where  /„  (z)  is  not  infinite  for  infinite  values  of  z.     Now  the  polynomial 

m 

Sttj^  is  not  infinite  for  any  finite  value  of  z  ;  hence  f0  (z)  is  infinite  for  all 

i  =  l 

the  finite  infinities  of  f  (z)  and  in  the  same  way,  that  is,  the  function  f0(z) 
has  «!,  ......  ,  a^  for  its  poles  and  it  has  no  other  singularities. 

Again,  since  «M  is  a  finite  pole  of  multiplicity  WM,  we  have 


where  fi(z)  is  not  infinite  for  z  =  all  and,  as  f0(z)  is  not  infinite  for  z=<x>  , 
evidently  f^  (z)  is  not  infinite  for  z  =  oo  .  Hence  the  singularities  of  f^  (z)  are 
merely  the  poles  a1}  ......  ,  aF_i  ;  and  these  are  all  its  singularities. 

Proceeding  in  this  manner  for  the  singularities  in  succession,  we  ultimately 
reach  a  function  f^  (z)  which  has  only  one  pole  a^  and  no  other  singularity, 
so  that 

k  k 


where  g  (z)  is  not  infinite  for  z  =  a^  But  the  function  f^(z)  is  infinite  only 
for  2  =  0,!,  and  therefore  g  (2)  has  no  infinity.  Hence  g  (z}  is  only  a  constant, 
say  k0  :  thus 

9  (*}  =  ^o- 

Combining  all  these  results  we  have  a,  finite  number  of  series  to  add  together: 
and  the  result  is  that 


72  UNIFORM  [48. 

where  g1  (z)  is  the  series  k0  +  a-^z  + +  amzm,  and       \  I  is  the  sum  of  the 

finite  number  of  fractions.     Evidently  gs  (z)  is  the  product 

{z  —  Oi)m>  (z  —  a2)ma (z  —  aM)mfx ; 

and  g»  (z)  is  at  most  of  degree 


If  F  (z}  denote  g1  (z}  g3  (z)  +  g^  (z),  the  form  of  /  (z)  is 


</.(*)' 

that  is,  f  (z)  is  a  rational,  algebraical,  meromorphic  function. 

It  is  evident  that,  when  the  function  is  thus  expressed  as  an  algebraical 
fraction,  the  degree  of  F  (z)  is  the  sum  of  the  multiplicities  of  all  the  poles 
when  infinity  is  a  pole. 

COROLLARY  I.  A  function,  all  the  singularities  of  which  are  accidental, 
has  as  many  zeros  as  it  has  accidental  singularities  in  the  plane. 

If  z  =  oo  be  a  pole,  then  it  follows  that,  because  f(z)  can  be  expressed 
in  the  form 


it  has  as  many  zeros  as  F(z),  unless  one  such  should  be  also  a  zero  of  g^(z). 
But  the  zeros  of  g3(z)  are  known,  and  no  one  of  them  is  a  zero  of  F(z),  on 
account  of  the  form  of  f(z}  when  it  is  expressed  in  partial  fractions.  Hence 
the  number  of  zeros  off(z)  is  equal  to  the  degree  of  F(z},  that  is,  it  is  equal 
to  the  number  of  poles  off(z}. 

If  2=00  be  not  a  pole,  two  cases  are  possible;  (i)  the  function  f  (z)  may  be 
finite  for  z  =  oo  ,  or  (ii)  it  may  be  zero  for  z  =  oo  .  In  the  former  case,  the 
number  of  zeros  is,  as  before,  equal  to  the  degree  of  F  (z),  that  is,  it  is  equal 
to  the  number  of  infinities. 

In  the  latter  case,  if  the  degree  of  the  numerator  F  (z)  be  K  less  than 
that  of  the  denominator  gs  (z),  then  z  =  oo  is  a  zero  of  multiplicity  K  ;  and  it 
follows  that  the  number  of  zeros  is  equal  to  the  degree  of  the  numerator 
together  with  K,  so  that  their  number  is  the  same  as  the  number  of  accidental 
singularities. 

COROLLARY  II.  At  the  beginning  of  the  proof  of  the  theorem  of  the 
present  section,  it  is  proved  that  a  function,  all  the  singularities  of  which  are 
accidental,  has  only  a  finite  number  of  such  singularities. 

Hence,  by  the  preceding  Corollary,  such  a  function  can  have  only  a  finite 
number  of  zeros. 

If,  therefore,  the  number  of  zeros  of  a  function  be  infinite,  the  function 
must  have  at  least  one  essential  singularity. 


48.]  ALGEBRAICAL   FUNCTIONS  73 

COROLLARY  III.  When  a  uniform  analytic  function  has  no  essential 
singularity,  if  the  (finite)  number  of  its  poles,  say  clv..,  cm,  be  m,  no  one  of 
them  being  at  z  =  oo ,  and  if  the  number  of  its  zeros,  say  aly...,  am,  be  also  m, 
no  one  of  them  being  at  z  =  oo  ,  then  the  function  is 


„ n 

*       a 


r=l  \Z  -  CT 

except  possibly  as  to  a  constant  factor. 

When  z  =  oo  is  a  zero  of  order  n,  so  that  the  function  has  m  —  n  zeros,  say 
«i,  a2,...,  in  the  finite  part  of  the  plane,  the  form  of  the  function  is 


m-n 

II  (z  —  ar) 

r=l 


r=l 


and,  when  z  =  <x>  is  a  pole  of  order  p,  so  that  the  function  has  m  -  p  poles, 
say  cl}  c.2>...,  in  the  finite  part  of  the  plane,  the  form  of  the  function  is 


II  (Z  -  Or) 

r=l  _ 

m-p  ~ 


COROLLARY  IV.     All  the  singularities  of  rational  algebraical  meromorphic 
functions  are  accidental. 


CHAPTER  V. 

TRANSCENDENTAL  INTEGRAL  FUNCTIONS. 

49.  WE  now  proceed  to  consider  the  properties  of  uniform  functions 
which  have  essential  singularities. 

The  simplest  instance  of  the  occurrence  of  such  a  function  has  already 
been  referred  to  in  §  42  ;  the  function  has  no  singularity  except  at  z  =  oo , 
and  that  value  is  an  essential  singularity  solely  through  the  failure  of  the 
limitation  to  finiteness  that  would  render  the  singularity  accidental.  The 
function  is  then  an  integral  function  of  transcendental  character ;  and  it  is 
analytically  represented  (§  26)  by  G  (z)  an  infinite  series  in  positive  powers  of 
z,  which  converges  everywhere  in  the  finite  part  of  the  plane  and  acquires 
an  infinite  value  at  infinity  alone. 

The  preceding  investigations  shew  that  uniform  functions,  all  the  singu 
larities  of  which  are  accidental,  are  rational  algebraical  functions — their 
character  being  completely  determined  by  their  uniformity  and  the  accidental 
nature  of  their  singularities,  and  that  among  such  functions  having  the  same 
accidental  singularities  the  discrimination  is  made,  save  as  to  a  constant 
factor,  by  means  of  their  zeros. 

Hence  the  zeros  and  the  accidental  singularities  of  a  rational  algebraical 
function  determine,  save  as  to  a  constant  factor,  an  expression  of  the  function 
which  is  valid  for  the  whole  plane.  A  question  therefore  arises  how  far 
the  zeros  and  the  singularities  of  a  transcendental  function  determine  the 
analytical  expression  of  the  function  for  the  whole  plane. 

50.  We  shall  consider  first  how  far  the  discrimination  of  transcendental 
integral  functions,  which  have  no  infinite  value  except  for  z  =  oc  ,  is  effected 
by  means  of  their  zeros*. 

*  The  following  investigations  are  based  upon  the  famous  memoir  by  Weierstrass,  "  Zur 
Theorie  der  eindeutigen  analytischen  Functionen,"  published  in  187G :  it  is  included,  pp.  1 — 52, 
in  the  Abhandlungen  aus  der  Functioiienlehre  (Berlin,  1886). 

In  connection  with  the  product-expression  of  a  transcendental  function,  Cayley,  "  Memoire  sur 
les  fonctions  doublement  periodiques,"  Liouville,  t.  x,  (1845),  pp.  385 — 420,  or  Collected  Works, 
vol.  i,  pp.  156 — 182,  should  be  consulted. 


50.] 


CONVERGING   INFINITE   PRODUCTS 


75 


Let  the  zeros  aly  a2,  a3,...  be  arranged  in  order  of  increasing  moduli;  a 
finite  number  of  terms  in  the  series  may  have  the  same  value  so  as  to  allow 
for  the  existence  of  a  multiple  zero  at  any  point.  After  the  results  stated 
47,  it  will  be  assumed  that  the  number  of  zeros  is  infinite  ;  that, 


n 


subject  to  limited  repetition,  they  are  isolated  points  ;  and,  in  the  present 
chapter,  that,  as  n  increases  indefinitely,  the  limit  of  \an\  is  infinity.  And  it 
will  be  assumed  that  at\  >  0,  so  that  the  origin  is  temporarily  excluded  from 
the  series  of  zeros. 

Let  z  be  any  point  in  the  finite  part  of  the  plane.  Then  only  a  limited 
number  of  the  zeros  can  lie  within  and  on  a  circle  centre  the  origin  and 
radius  equal  to  \z\  ;  let  these  be  a]5  a2,...,  afc_1}  and  let  ar  denote  any  one  of 
the  other  zeros.  We  proceed  to  form  the  infinite  product  of  quantities  ur, 
where  ur  denotes 


and  gr  is  a  rational  integral  function  of  z  which,  being  subject  to  choice,  will 
be  chosen  so  as  to  make  the  infinite  product  converge  everywhere  in  the 
plane.  We  have 

00        \ 


w=l 

a  series  which  converges  because  \z  <  \ar\.     Now  let 

ffr  = 

then 


«>    1   /  ^  \n 
i  v    -1-  /  •*  \ 

logi<r  =  -  2  -f£J   , 

»j  =  S  »4  \**rr 


and  therefore 


Hence 


•-— " 


if  the  expression  on  the  right-hand  side  be  finite,  that  is,  if  the  series 

oo       ce     I     /  _  \  n 

2   S  -(-) 

r=ftw=«^  \flrf 

converge  unconditionally.     Denoting  the  modulus  of  this  series  by  M,  we 
have 

z 
a,. 


00  00        1 

M  <  2    2  - 

r-k n=s M 


SO  that 


sM<  S    2 

r=k n=s 


7G 


WEIERSTRASS'S   CONVERGING 


[50. 


whence  since  1  -  —   is  the  smallest  of  the  denominators  in  terms  of  the  last 

«* 
sum,  we  have 


sM\l- 

z 

[  < 

00 

Z 

8 

1 

«& 

j        r=k 

ar 

•  I  l 

• 

*-l 

If,  as  is  not  infrequently  the  case,  there  be  any  finite  integer  s  for  which  (and 
therefore  for  all  greater  indices)  the  series 

2     1 


Is  ' 


00 

and  therefore  the  series    2  \ar\-s,  converges,  we  choose  s  to  be  that  least 

r=k 

integer.     The  value  of  M  then  is  finite  for  all  finite  values  of  z  ;  the  series 


oo        co     T    /  ~\n 

2   2  -  - 


n 

r=k 


converges  unconditionally  and  therefore 
is  a  converging  product  when 

Let  the  finite  product 

A-l    (/  f 

n  |(i-- 

m=l  l\          am 

be  associated  as  a  factor  with  the  foregoing  infinite  converging  product.   Then 
the  expression 

oo     ( f  2  \       2 

T-=I  (\        ar/ 
is  an  infinite  product,  converging  uniformly  and  unconditionally  for  all  finite 

00 

values  of  z,  provided  the  finite  integer  s  be  such  as  to  make  the  series  2 


converge  uniformly  and  unconditionally. 

Since  the  product  converges  uniformly  and  unconditionally,  no  product 
constructed  from  its  factors  ur,  say  from  all  but  one  of  them,  can  be  infinite. 
Now  the  factor 

"5?i/£-Y 

\ ?L\en=\n\am) 


vanishes  for  z  =  am;  hence  f(z)  vanishes  for  z  =  am.  Thus  the  function, 
evidently  uniform  after  what  has  been  proved,  has  the  assigned  points 
Oj,  a2)...  and  no  others  for  its  zeros. 


50.] 


INFINITE   PRODUCT 


77 


Further,  z  =  oo  is  an  essential  singularity  of  the  function  ;  for  it  is  an 
essential  singularity  of  each  of  the  factors  on  account  of  the  exponential 
element  in  the  factor. 

51.  But  it  may  happen  that  no  finite  integer  s  can  be  found  which  will 
make  the  series 

00 

r=l 

converge*.     We  then  proceed  as  follows. 

Instead  of  having  the  same  index  s  throughout  the  series,  we  associate 
with  every  zero  ar  an  integer  mr  chosen  so  as  to  make  the  series 


n=l    @"n  \Q"n 

a  converging  series.  To  obtain  these  integers,  we  take  any  series  of  decreasing 
real  positive  quantities  e,  e1}  e2,...,  such  that  (i)  e  is  less  than  unity  and 
(ii)  they  form  an  unconditionally  converging  series ;  and  we  choose  integers 
ftir  such  that 


These  integers  make  the  foregoing  series  of  moduli  converge.  For, 
neglecting  the  limited  number  of  terms  for  which  \z\^  a\,  and  taking  e 
such  that 

z 


we  have  for  all  succeeding  terms 


and  therefore 


ar 


Hence,  except  for  the  first  k  —  1  terms,  the  sum  of  which  is  finite,  we  have 


n=k 


which  is  finite  because  the  series 


...  converges.     Hence  the  series 


n=l 


s  a  converging  series. 


*  For  instance,  there  is  no  finite  integer  s  that  can  make  the  infinite  series 
(log  2)-'  +  (log  3)-  +  (log  4)-  +  .  .  . 

converge.     This  series  is  given  in  illustration  by  Hermite,  Cours  a  la  faculte  des  Sciences  (4mc  ed. 
1891),  p.  86. 


78 


WEIERSTRASS'S   CONVERGING 


[51. 


Just  as  in  the  preceding  case  a  special  expression  was  formed  to  serve  as 
a  typical  factor  in  the  infinite  product,  we  now  form  a  similar  expression 
for  the  same  purpose.  Evidently 


1  -  a;  =  ei<*  a-*)  =  e 
if  \x\  <  1.     Forming  a  function  E  (x,  m)  denned  by  the  equation 


m   xr 

S  - 


E  (x,  m)=(l-x)e r=1  r  , 


we  have  E  (x,  m)  = 

In  the  preceding  case  it  was  possible  to  choose  the  integer  m  so  that  it 
should  be  the  same  for  all  the  factors  of  the  infinite  product,  which    was 

0 

ultimately  proved  to  converge.     Now,  we  take  x  =  —   and  associate  mn  as 
the  corresponding  value  of  m.     Hence,  if 

/(*)  = 


where 


<  \z  <  |ttjfc|,  we  have 


n=k 


-  s   s 


The  infinite  product  represented  by  f(z)  will  converge  if  the  double  series  in 
the  exponential  be  a  converging  series. 

Denoting  the  double  series  by  S,  we  have 


\S\<*    2 


2^* 
2 

n=kr=l 


r+mn 


<  2 

n—k 


1+TOM 


1 4£ 

\an 


on  effecting  the  summation  for  r.     Let  A  be  the  value  of  1  — 


all  the  remaining  values  of  n  we  have 

1 


z  ! 


-   ;  then  for 


—  >>A, 


and  so 


n=/fc 


This  series  converges;   hence  for  finite  values  of   z\  the  value  of  \S\  is 
finite,  so  that  S  is  a  converging  series.     Hence  it  follows  that  f(z)  is  an 


51.]  INFINITE    PRODUCT  79 

unconditionally  converging  product.     We  now  associate  with  f(z)  as  factors 
the  k  —  I  functions 


for  i=  1,  2,...,  k—1;  their  number  being  finite,  their  product  is  finite  and 
therefore  the  modified  infinite  product  still  converges.     We  thus  have 


an  unconditionally  converging  product. 

Since  the  product  G  (z)  converges  unconditionally,  no  product  constructed 
from  its  factors  E,  say  from  all  but  one  of  them,  can  be  infinite.     The  factor 


vanishes  for  the  value  z  =  an  and  only  for  this  value  ;  hence  G  (z)  vanishes  for 
z  =  an.  It  therefore  appears  that  G(z)  has  the  assigned  points  a1}  a.,,  a3,  ... 
and  no  others  for  its  zeros ;  and  from  the  existence  of  the  exponential  in  each 
of  the  factors  it  follows  that  z  =  oo  is  an  essential  singularity  of  the  factor  and 
therefore  it  is  an  essential  singularity  of  the  function. 

Denoting  the  series  in  the  exponential  by  gn  (z\  so  that 

mn    1    /  ~  \  r 

*<*>-£?(£)• 

71  /  z  \         i-.  Z\ 

we  have  A    — ,  mn    =    1 e^'  ; 

\an        /      V        aJ 

and  therefore  the  function  obtained  is 

;       G (z)=  H  \(l  —  —  ]  eg«(zl 

n  =  l  (\  Q"n,l 

The  series  gn  usually  contains  only  a  limited  number  of  terms ;  when  the 
number  of  terms  increases  without  limit,  it  is  only  with  indefinite  increase 
of  |  an  |  and  the  series  is  then  a  converging  series. 

It  should  be  noted  that  the  factors  of  the  infinite  product  G  (z)  are  the 
expressions  E  no  one  of  which,  for  the  purposes  of  the  product,  is  resoluble 
into  factors  that  can  be  distributed  and  recombined  with  similarly  obtained 
factors  from  other  expressions  E;  there  is  no  guarantee  that  the  product 
of  the  factors,  if  so  resolved,  would  converge  uniformly  and  unconditionally, 
and  it  is  to  secure  such  convergence  that  the  expressions  E  have  been 
constructed. 

It  was  assumed,  merely  for  temporary  convenience,  that  the  origin  was  not 
a  zero  of  the  required  function ;  there  obviously  could  not  be  a  factor  of 
exactly  the  same  form  as  the  factors  E  if  a  were  the  origin. 


80  TRANSCENDENTAL    INTEGRAL    FUNCTION  [51. 

If,  however,  the  origin  were  a  zero  of  order  X,  we  should  have  merely 
to  associate  a  factor  ZK  with  the  function  already  constructed. 

We  thus  obtain  Weierstrass's  theorem : — 

It  is  possible  to  construct  a  transcendental  integral  function  such  that  it 
shall  have  infinity  as  its  only  essential  singularity  and  have  the  origin  (of 
multiplicity  X),  a^,  az,  a3,  ...  as  zeros ;  and  such  a  function  is 

00    ( /          z\ 

ZK  n  ui  —  U^ 


n=i 

where  gn(z)  is  a  rational,  integral,  algebraical  function  of  z,  the  form  of  which 
is  dependent  upon  the  law  of  succession  of  the  zeros. 

52.  But,  unlike  uniform  functions  with  only  accidental  singularities,  the 
function  is  not  unique  :  there  are  an  unlimited  number  of  transcendental 
integral  functions  with  the  same  series  of  zeros  and  infinity  as  the  sole  essential 
singularity,  a  theorem  also  due  to  Weierstrass. 

For,  if  G!  (z)  and  G  (z)  be  two  transcendental,  integral  functions  with  the 
same  series  of  zeros  in  the  same  multiplicity,  and  z  =  oo  as  their  only  essential 
singularity,  then 


G(z} 
is  a  function  with  no  zeros  and  no  infinities  in  the  finite  part  of  the  plane. 

Denoting  it  by  £r2,  then 

1  ^ 

<72  dz 

is  a  function  which,  in  the  finite  part  of  the  plane,  has  no  infinities;  and 
therefore  it  can  be  expanded  in  the  form 


a  series  converging  everywhere  in  the  finite  part  of  the  plane.     Choosing  a 
constant  C0  so  that  6r2  (0)  =  e*7",  we  have  on  integration 


where  g(z)  =  C0 

and  g  (z)  is  finite  everywhere  in  the  finite  part  of  the  plane.  Hence  it  follows 
that,  ifg(z)  denote  any  integral  function  of  z  which  is  finite  everywhere  in  the 
finite  part  of  the  plane,  and  if  G  (z)  be  some  transcendental  integral  function 
with  a  given  series  of  zeros  and  z=  oo  as  its  sole  essential  singularity,  all 
transcendental  integral  functions  with  that  series  of  zeros  and  z=  <x>  as  the 
sole  essential  singularity  are  included  in  the  form 

£(*)«*». 

COROLLARY  I.  A  function  which  has  no  zeros  in  the  finite  part  of  the 
plane,  no  accidental  singularities  and  z=<x>  for  its  sole  essential  singularity 
is  necessarily  of  the  form 


52.]  AS   AN    INFINITE   PRODUCT  81 

where  g  (z)  is  an  integral  function  of  z  finite  everywhere  in  the  finite  part 
of  the  plane. 

COROLLARY  II.  Every  transcendental  function,  which  has  the  same  zeros 
in  the  same  multiplicity  as  an  algebraical  polynomial  A  (z)  —  the  number, 
therefore,  being  necessarily  finite  —  ,  ivhich  has  no  accidental  singularities  and 
has  z  =  oo  for  its  sole  essential  singularity,  can  be  expressed  in  the  form 

A  (z) 


COROLLARY  III.  Every  function,  which  has  an  assigned  series  of  zeros 
and  an  assigned  series  of  poles  and  has  z  =  oo  for  its  sole  essential  singu 
larity,  is  of  the  form 


where  the  zeros  of  G0(z)  are  the  assigned  zeros  and  the  zeros  of  Gp(z)  are  the 
assigned  poles. 

For  if  Op  (z)  be  any  transcendental  integral  function,  constructed  as  in 
the  proposition,  which  has  as  its  zeros  the  poles  of  the  required  function  in 
the  assigned  multiplicity,  the  most  general  form  of  that  function  is 

0p(*)e*», 

where  h  (z)  is  integral.  Hence,  if  the  most  general  form  of  function  which 
has  those  zeros  for  its  poles  be  denoted  by  f(z),  we  have 

f(z)Gp(z)e^ 

as  a  function  with  no  poles,  with  infinity  as  its  sole  essential  singularity,  and 
with  the  assigned  series  of  zeros.  But  if  G0  (z)  be  any  transcendental  integral 
function  with  the  assigned  zeros  as  its  zeros,  the  most  general  form  of  function 
with  those  zeros  is 


and  so  f(z)  Gp  (z)  eh  ®  =  G0  (z)  e°  &  , 

whence  /  (z)  =  ?$1  effW, 

Lrp  (z) 

in  which  g  (z)  denotes  g  (z)  —  h  (z). 

If  the  number  of  zeros  be  finite,  we  evidently  may  take  G0(z)  as  the 
algebraical  polynomial  with  those  zeros  as  its  only  zeros. 

If  the  number  of  poles  be  finite,  we  evidently  may  take  Gp(z)  as  the 
algebraical  polynomial  with  those  poles  as  its  only  zeros. 

And,  lastly,  if  a  function  have  a  finite  number  of  zeros,  a  finite  number 
of  accidental  singularities  and  2=00  as  its  sole  essential  singularity,  it  can 
be  expressed  in  the  form 


F. 


82  PRIMARY  [52. 

where  P  and  Q  are  rational  integral  polynomials.  This  is  valid  even  though 
the  number  of  assigned  zeros  be  not  the  same  as  the  number  of  assigned 
poles  ;  the  sole  effect  of  the  inequality  of  these  numbers  is  to  complicate  the 
character  of  the  essential  singularity  at  infinity. 

53.  It  follows  from  what  has  been  proved  that  any  uniform  function, 
having  z  =  <x>  for  its  sole  essential  singularity  and  any  number  of  assigned 
zeros,  can  be  expressed  as  a  product  of  expressions  of  the  form 


a 


Such  a  quantity  is  called*  a  primary  factor  of  the  function. 

It  has  also  been  proved  that : — 

(i)     If  there  be  no  zero  an,  the  primary  factor  has  the  form 

(ii)  The  exponential  index  gn  (z)  may  be  zero  for  individual  primary 
factors,  though  the  number  of  such  factors  must,  at  the  utmost, 
be  finite  f. 

(iii)     The  factor  takes  the  form  z  when  the  origin  is  a  zero. 
Hence  we  have  the  theorem,  due  to  Weierstrass : — 

Every  uniform  integral  function  of  z  can  be  expressed  as  a  product  of 
primary  factors,  each  of  the  form 

(kz  +  I)  e3W, 

where  g(z)  is  an  appropriate  integral  function  of  z  vanishing  with  z  and  where 
k,  I  are  constants.  In  particular  factors,  g  (z)  may  vanish ;  and  either  k  or  I, 
but  not  both  k  and  I,  may  vanish  with  or  without  a  non-vanishing  exponential 
index  g(z). 

54.  It  thus  appears  that  an  essential  distinction  between  transcendental 
integral  functions  is  constituted  by  the  aggregate  of  their  zeros :  and  we  may 
conveniently  consider  that  all  such  functions  are  substantially  the  same  when 
they  have  the  same  zeros. 

There  are  a  few  very  simple  sets  of  functions,  thus  discriminated  by  their 
zeros:  of  each  set  only  one  member  will  be  given,  and  the  factor  e^(z},  which 
makes  the  variation  among  the  members  of  the  same  set,  will  be  neglected 
for  the  present.  Moreover,  it  will  be  assumed  that  the  zeros  are  isolated 
points. 

I.  There  may  be  a  finite  number  of  zeros ;  the  simplest  function  is  then 
an  algebraical  polynomial. 

*  Weierstrass's  term  is  Prim/unction,  I.e.,  p.  15. 

t  Unless  the  class  (§  59)  be  zero,  when  the  index  is  zero  for  all  the  factors. 


54.]  FACTORS  83 

II.  There  may  be  a  singly-infinite  system  of  zeros.  Various  functions 
will  be  obtained,  according  to  the  law  of  distribution  of  the  zeros. 

Thus  let  them  be  distributed  according  to  a  law  of  simple  arithmetic 
progression  along  a  given  line.  If  a  be  a  zero,  co  a  quantity  such  that  co  \ 
is  the  distance  between  two  zeros  and  arg.  co  is  the  inclination  of  the  line, 
we  have 

a  +  mco, 

for  integer  values  of  m  from  -  oo  to  +  oo  ,  as  the  expression  of  the  series  of 
the  zeros.  Without  loss  of  generality  we  may  take  a  at  the  origin  —  this 
is  merely  a  change  of  origin  of  coordinates  —  and  the  origin  is  then  a 
simple  zero  :  the  zeros  are  given  by  mco,  for  integer  values  of  m  from 
—  oo  to  +  oo  . 

Now  2  —  -  =  -  2  —  is  a  diverging  series  ;  but  an  integer  s  —  the  lowest 

value  is  s  =  2  —  can  be  found  for  which  the  series   S  I  -  ]   converges  uni- 

\mcoj 
formly  and  unconditionally.     Taking  s  =  2,  we  have 

,  .      '-1  1  /  z  \n       z 
ffm  (z)  =  2  -    —     =  —  , 
»=i  n  vW       m™ 
so  that  the  primary  factor  of  the  present  function  is 


Z   \ 

---  ) 
mco/ 


m<a 

e 


and  therefore,  by  §  52,  the  product 

/«-,SJ(i-  *-) 

-oo  (\        mcoj 
converges  uniformly  and  unconditionally  for  all  finite  values  of  z. 

The  term  corresponding  to  m  =  0  is  to  be  omitted  from  the  product  ;  and 
it  is  unnecessary  to  assume  that  the  numerical  value  of  the  positive  infinity 
for  m  is  the  same  as  that  of  the  negative  infinity  for  m.  If,  however,  the 
latter  assumption  be  adopted,  the  expression  can  be  changed  into  the  ordinary 
product-expression  for  a  sine,  by  combining  the  primary  factors  due  to  values 
of  m  that  arc  equal  and  opposite  :  in  fact,  then 


co    .    TTZ 
=  -  -  sin  — . 

7T  CO 


This  example  is  sufficient  to  shew  the  importance  of  the  exponential  term  in  the 
primary  factor.  If  the  product  be  formed  exactly  as  for  an  algebraical  polynomial,  then 
the  function  is 


z  n 

in  the  limit  when  both  p  and  q  are  infinite.     But  this  is  known*  to  be 


-  )    -  sin  — . 

77  0) 


*  Hobson's  Trigonometry,  §  287. 

6—2 


84  PRIMARY  [54. 

Another  illustration  is  afforded  by  Gauss's  II-function,  which  is  the  limit  when  k  is 
infinite  of 

1.2.3  ......  k 

(«+!)  (0+2)  ......  (z+k) 

This  is  transformed  by  Gauss*  into  the  reciprocal  of  the  expression 


that  is,  of  (1  +*)  jj  {(l  +^)  e  "2l°g 

the   primary  factors  of  which  have   the  same   characteristic  form   as   in   the  preceding 
investigation,  though  not  the  same  literal  form. 

It  is  chiefly  for  convenience  that  the  index  of  the  exponential  part  of  the  primary 

t-l   1/2  \n 

factor  is  taken,  in  §  50,  in  the  form    2   -  (  —  )   .      With  equal  effectiveness  it  may  be 

n=l  %    \~^T  / 
»-l  1 

taken  in  the  form   2  -  br  nzn.  provided  the  series 
' 


r=k  «=i   n 
converge  uniformly  and  unconditionally. 

Ex.  1.     Prove  that  each  of  the  products 


form=+l,  ±3,  +5,  ......  to  infinity,  and 


the  term  for  n  =  Q  being  excluded  from  the  latter  product,  converges  uniformly  and  uncon 
ditionally  and  that  each  of  them  is  equal  to  cos  z.  (Hermite  and  Weyr.) 

Ex.  2.  Prove  that,  if  the  zeros  of  a  transcendental  integral  function  be  given  by  the 
series 

0)   +&>,   ±4w,   +9cB,  ......  to  infinity, 

the  simplest  of  the  set  of  functions  thereby  determined  can  be  expressed  in  the  form 

(    fz\*\    ,     (.    fz\*\ 
sm  X?r  I  -     }-  sin  -UTT     -  )  }-  . 
I     W  )          (      W  J 

Ex.  3.  Construct  the  set  of  transcendental  integral  functions  which  have  in  common 
the  scries  of  zeros  determined  by  the  law  m2a>l  +  2m<a2  +  a>3  for  all  integral  values  of  m 
between  -  oo  and  +  oo  ;  and  express  the  simplest  of  the  set  in  terms  of  circular  functions,  j 

55.  The  law  of  distribution  of  the  zeros,  next  in  importance  and  sub 
stantially  next  in  point  of  simplicity,  is  that  in  which  the  zeros  form  a  doubly- 
infinite  double  arithmetic  progression,  the  points  being  the  oo  2  intersections 
of  one  infinite  system  of  equidistant  parallel  straight  lines  with  another 
infinite  system  of  equidistant  parallel  straight  lines. 

The  origin  may,  without  loss  of  generality,  be  taken  as  one  of  the  zeros. 
If  a)  be  the  coordinate  of  the  nearest  zero  along  the  line  of  one  system 
passing  through  the  origin,  and  &>'  be  the  coordinate  of  the  nearest  zero  along 

*  Ges.  Wcrke,  t.  Hi,  p.  145;   the  example  is  quoted  in  this  connection  by  Weierstrass,  I.e.,  ! 
p.  15. 


55.]  FACTIOUS  85 

the  line  of  the  other  system  passing  through  the  origin,  then  the  complete 
series  of  zeros  is  given  by 

fl  =  mw  +  mm, 

for  all  integral  values  of  m  and  all  integral  values  of  ni  between  —  <x>  and 
+  oo .  The  system  of  points  may  be  regarded  as  doubly -periodic,  having  &> 
arid  &>'  for  periods. 

It  must  be  assumed  that  the  two  systems  of  lines  intersect.  Other 
wise,  w  and  to'  would  have  the  same  argument  and  their  ratio  would  be  a  real 
quantity,  say  a ;  and  then 

ft 

—  =  m  +  m  a. 

CO 

Whether  a  be  commensurable  or  incommensurable,  the  number  of  pairs 
of  integers,  for  which  m  +  in' a.  is  zero  or  may  be  made  less  than  any  small 
quantity  8,  is  infinite ;  and  in  either  case  we  should  have  the  origin  a  zero 
for  each  such  pair,  that  is,  altogether  the  origin  would  be  a  zero  of  infinite 
multiplicity.  This  property  of  a  function  is  to  be  considered  as  excluded, 
for  it  would  make  the  origin  an  essential  singularity  instead  of,  as  required, 
an  ordinary  point  of  the  transcendental  integral  function.  Hence  the  ratio  of 
the  quantities  w  and  w'  is  not  real. 

56.  For  the  construction  of  the  primary  factor,  it  is  necessary  to  render 
the  series 


converging,    by   appropriate    choice    of  integers   sm>m.      It  is   found   to    be 
possible  to  choose  an  integer  s  to  be  the  same  for  every  term  of  the  series, 
corresponding  to  the  simpler  case  of  the  general  investigation,  given  in  §  50. 
As  a  matter  of  fact,  the  series 

diverges  for  s  =  I  (we  have  not  made  any  assumption  that  the  positive  and 
the  negative  infinities  for  m  are  numerically  equal,  nor  similarly  as  to  m') ; 
the  series  converges  for  s  =  2,  but  its  value  depends  upon  the  relative  values 
of  the  infinities  for  m  and  m';  and  s  =  3  is  the  lowest  integral  value  for  which, 
as  for  all  greater  values,  the  series  converges  uniformly  and  unconditionally. 

There  are  various  ways  of  proving  the  uniform  and  unconditional  conver 
gence  of  the  series  2ft~M  when  /*  >  2 :  the  following  proof  is  based  upon  a 
general  method  due  to  Eisenstein*. 


»I=QO      n=oo 


First,  the  series     S        2     (m2  +  n*)~*  converges  uniformly  and  uricondi- 

m=  —  «>  n=  -oo 

tionally,  if  /j,>  1.    Let  the  series  be  arranged  in  partial  series :  for  this  purpose, 


Crelle,  t.  xxxv,  (1847),  p.  161 ;  a  geometrical  exposition  is  given  by  Halphen,  Traite  des 
fonctions  elliptiques,  t.  i,  pp.  358 — 362. 


86  WEIERSTRASS'S   FUNCTION    AS  [56. 

we  choose   integers   k   and   I,  and   include  in  each   such  partial  series  all 
the  terms  which  satisfy  the  inequalities 

m  ^  2*+1, 


so  that  the  number  of  values  of  m  is  2*  and  the  number  of  values  of  n  is  2*. 
Then,  if  k  +  I  =  %K,  we  have 


so  that  each  term  in  the  partial  series  ^  ^-  .     The  number  of  terms  in  the 

^"   J* 

partial  series  is  2fc  .  2*,  that  is,  22K  :  so  that  the  sum  of  the  terms  in  the 
partial  series  is 


Take  the  upper  limit  of  k  and  I  to  be  p,  ultimately  to  be  made  infinite. 
Then  the  sum  of  all  the  partial  series  is 


which,  when  p  =  oo  ,  is  a  finite  quantity  if  p  >  1. 
Next,  let  (a  =  a.  +  /3i,  «'  =  7  +  Si,  so  that 

ft  =  mw  +  nay'  =  ma  +  ny  +  i  (m{3  +  n8)  ; 
hence,  if  6  =•  ma.  +  nj,    (j>  =  m(3  +  n$, 

we  have  |  ft  2  =  fr  +  </>2. 

Now  take  integers  r  and  s  such  that 

r<0<r  +  \,    s<(jxs  +  ~L. 

The  number  of  terms  ft  satisfying  these  conditions  is  definitely  finite  and  is 
independent  of  m  and  n.     For  since 

m(«S  — 


n  a   - 

and  a8  —  (3y  does  not  vanish  because  o>'/a>  is  not  purely  real,  the  number  of 
values  of  in  is  the  integral  part  of 

(r  +  1)8  —  sy 

a.8  —  fiy 
less  the  integral  part  of 

r8  —  (s  +  1 )  7 
a.8  —  fly 

that  is,  it  is  the  integral  part  of  (7  +  8)/(«8  —  #7).     Similarly,  the  number  of 
values  of  n  is  the  integral  part  of  (a  +  /3)/(aS  -  j3j).     Let  the  product  of  the 


56.]  A   DOUBLY-INFINITE   PRODUCT  87 

last  two  integers  be  q  ;   then   the  number   of  terms  fl  satisfying  the  in 
equalities  is  q. 

Then  22  1  ft  \~*  =  22  (&>  +  p)~* 

<  q  22  (r2  +  s2)-'*, 
which,  by  the  preceding  result,  is  finite  when  yu,>  1.     Hence 

22  (mco  +  m'(»)'}~-»- 

converges  uniformly  and  unconditionally  when  //,  >  1  ;  and  therefore  the  least 
value  of  s,  an  integer  for  which 

22  (mco  +  m'co')~s 
converges  uniformly  and  unconditionally,  is  3. 

The  series  22(?tto)  +  m'<»')~2  has  a  finite  sum,  the  value  of  which  depends*  upon 
the  infinite  limits  for  the  summation  with  regard  to  m  and  m'.  This  dependence  is 
inconvenient  and  it  is  therefore  excluded  in  view  of  our  present  purpose. 

Ex.     Prove  in  the  same  manner  that  the  series 


the  multiple  summation  extending  over  all  integers  mlt  m2,  ......  ,  mn  between   —  oo  and 

+  oo  ,  converges  uniformly  and  unconditionally  if  2/j.>n.  (Eiseustein.) 

57.  Returning  now  to  the  construction  of  the  transcendental  integral 
function  the  zeros  of  which  are  the  various  points  H,  we  use  the  preceding 
result  in  connection  with  §  50  to  form  the  general  primary  factor.  Since 
s  =  3,  we  have 

s-l 


and  therefore  the  primary  factor  is 


Moreover,  the  origin  is  a  simple  zero.  Hence,  denoting  the  required  function 
by  a  (z),  we  have 

00       °° 

<r(z)  =  zU    H 

—  00     -00 

as  a  transcendental  integral  function  which,  since  the  product  converges  uni 
formly  and  unconditionally  for  all  finite  values  of  z,  exists  and  has  a  finite 
value  everywhere  in  the  finite  part  of  the  plane;  the  quantity  O  denotes 
mco  +  mV,  and  the  double  product  is  taken  for  all  values  of  m  and  of  m 
between  —  oo  and  +  oo  ,  simultaneous  zero  values  alone  being  excluded. 

This  function  will  be  called  Weierstrass's  o-function ;  it  is  of  importance 
in  the  theory  of  doubly-periodic  functions  which  will  be  discussed  in  Chapter 
XL 

*  See  a  paper  by  the  author,  Quart.  Journ.  of  Math.,  vol.  xxi,  (1886),  pp.  261—280. 


88  PRIMARY    FACTORS  [57. 

Ex.     If  the  doubly-infinite  series  of  zeros  be  the  points  given  by 

Q  =  m2^  +  2wm&>2  +  «2o>3, 

wi>  W2)  W3  being  such  complex  constants  that  i2  does  not  vanish  for  real  values  of  m  and  n, 
then  the  series 

2   2  Q-* 

converges  for  s  =  2.     The  primary  factor  is  thus 


and  the  simplest  transcendental  integral  function  having  the  assigned  zeros  is 


The  actual  points  that  are  the  zeros  are  the  intersections  of  two  infinite  systems  of 
parabolas. 

58.  One  more  result  —  of  a  negative  character  —  will  be  adduced  in  this 
connection.  We  have  dealt  with  the  case  in  which  the  system  of  zeros  is  a 
singly-infinite  arithmetical  progression  of  points  along  one  straight  line  and 
with  the  case  in  which  the  system  of  zeros  is  a  doubly-infinite  arithmetical 
progression  of  points  along  two  different  straight  lines  :  it  is  easy  to  see  that 
a  uniform  transcendental  integral  function  cannot  exist  with  a  triply  -infinite 
arithmetical  progression  of  points  for  zeros. 

A  triply-infinite  arithmetical  progression  of  points  would  be  represented 
by  all  the  possible  values  of 


for  all  possible  integer  values  for  p1}  p.,,  p3  between  —  oo  and  +  oc  ,  where  no 
two  of  the  arguments  of  the  complex  constants  flj,  H2,  O3  are  equal.  Let 

tlr  =  o)r  +  i(or',    (r  =  1,  2,  3)  ; 

then,  as  will  be  proved  (§  107)  in  connection  with  a  later  proposition,  it  is 
possible*  —  and  possible  in  an  unlimited  number  of  ways  —  to  determine 
integers  plt  p-2,ps  so  that,  save  as  to  infinitesimal  quantities, 

Pi          _  _  £2  ___  PS 


all  the  denominators  in  which  equations  differ  from  zero  on  account  of  the 
fact  that  no  two  arguments  of  the  three  quantities  fl1}  H2,  Ha  are  equal.  For 
each  such  set  of  determined  integers  we  have 

&.Qi+p&+p»to» 

zero  or  infinitesimal,  so  that  the  origin  is  a  zero  of  unlimited  multiplicity  or, 
in  other  words,  there  is  a  space  at  the  origin  containing  an  unlimited  number 
of  zeros.  In  either  case  the  origin  is  an  essential  singularity,  contrary  to 

*  Jacobi,  Oes.  Werke,  t.  ii,  p.  27. 


58.]  CLASS   OF   A   FUNCTION  89 

the  hypothesis  that  the  only  essential  singularity  is  for  z  —  oo  ;  and  hence  a 
uniform  transcendental  function  cannot  exist  having  a  triply-infinite  arith 
metical  succession  of  zeros. 

59.  In  effecting  the  formation  of  a  transcendental  integral  function  by 
means  of  its  primary  factors,  it  was  seen  that  the  expression  of  the  primary 
factor  depends  upon  the  values  of  the  integers  which  make 


a  converging  series.  Moreover,  the  primary  factors  are  not  unique  in  form, 
because  any  finite  number  of  terms  of  the  proper  form  can  be  added  to  the 
exponential  index  in 


and  such  terms  will  only  the  more  effectively  secure  the  convergence  of  the 
infinite  product.  But  there  is  a  lower  limit  to  the  removal  of  terms  with  the 
highest  exponents  from  the  index  of  the  exponential  ;  for  there  are,  in  general, 
minimum  values  for  the  integers  m1}  m»,...,  below  which  these  integers  can 
not  be  reduced,  if  the  convergence  of  the  product  is  to  be  secured. 

The  simplest  case,  in  which  the  exponential  must  be  retained  in  the 
primary  factor  in  order  to  secure  the  convergence  of  the  infinite  product,  is 
that  discussed  in  §  50,  viz.,  when  the  integers  ml,  w2)...  are  equal  to  one 
another.  Let  m  denote  this  common  value  for  a  given  function,  and  let 
m  be  the  least  integer  effective  for  the  purpose  :  the  function  is  then  said* 
to  be  of  class  m,  and  the  condition  that  it  should  be  of  class  m  is  that  the 
integer  m  be  the  least  integer  to  make  the  series 


converge  uniformly  and  unconditionally,  the  constants  a  being  the  zeros  of 
the  function. 

Thus  algebraical  polynomials  are  of  class  0  ;  the  circular  functions  sin  z 
and  cos  z  are  of  class  1  ;  Wcierstrass's  o--function,  and  the  Jacobian  elliptic 
function  sn  z  are  of  class  2,  and  so  on  :  but  in  .no  one  of  these  classes  do  the 
functions  mentioned  constitute  the  whole  of  the  functions  of  that  class. 

60.  One  or  two  of  the  simpler  properties  of  an  aggregate  of  transcen 
dental  integral  functions  of  the  same  class  can  easily  be  obtained. 

Let  a  function  f(z),  of  class  n,  have  a  zero  of  order  r  at  the  origin  and 

*  The  French  word  is  genre  ;  the  Italian  is  genere.    Laguerre  (see  references  on  p.  92)  appears 
to  have  been  the  first  to  discuss  the  class  of  transcendental  integral  functions. 


90 


CLASS-PROPERTIES   OF 


[60. 


have  «!,  a2)...  for  its  other  zeros,  arranged  in  order  of  increasing  moduli. 
Then,  by  §  50,  the  function /O)  can  be  expressed  in  the  form 


(*)=' 


M    1  /  £\8 

where  </;  (V)  denotes  the  series  2  -f— 1    and  G(z)  must  be  properly  deter 
mined  to  secure  the  equality. 

Now  the  series 


is  one  which  converges  uniformly  for  all  values  of  z  that  do  not  coincide  with 
one  of  the  points  a,  that  is,  with  one  of  the  zeros  of  the  original  function. 
For  the  sum  of  the  series  of  the  moduli  of  its  terms  is 


1 


Let  d  be  the  least  of  the  quantities 


1 


,  necessarily  non-evanescent  be 


cause  z  does  not  coincide  with  any  of  the  points  a ;  then  the  sum  of  the  series 

IS       1 


which  is  a  converging  series  since  the  function  is  of  class  n.  Hence  the 
series  of  moduli  converges  and  therefore  the  original  series  converges  ;  let  it 
be  denoted  by  S  (z),  so  that 

1 


=2 


We  have 


Each  step  of  this  process  is  reversible  in  all  cases  in  which  the  original  pro- 

f  (z\ 
duct  converges;  if,  therefore,  it  can  be  shewn  of  a  function  f(z)  that  -rr4 

takes  this  form,  the  function  is  thereby  proved  to  be  of  class  n. 
If  there  be  no  zero  at  the  origin,  the  term  -  is  absent. 


CO.]  TRANSCENDENTAL  INTEGRAL  JUNCTIONS  91 

If  the  exponential  factor  G(z)  be  a  constant  so  that  G' (z)  is  zero,  the 
function /(.z)  is  said  to  be  a  simple  function  of  class  n. 

61.  There  are  one  or  two  criteria  to  determine  the  class  of  a  function : 
the  simplest  of  them  is  contained  in  the  following  proposition,  due  to 
Laguerre*. 

If,  as  z  tends  to  the  value  <x>  ,  a  very  great  value  of    z    can  be  found  for 

f'(z\ 
which  the  limit  of  z~n  --jr\  ,  where  f  (z)  is  a  transcendental,  integral  function, 

J\z) 
tends  uniformly  to  the  value  zero,  then  f  (z}  is  of  class  n. 

Take  a  circle  centre  the  origin  and  radius  R,  equal  to  this  value  of  \z\\ 
then,  by  §  24,  II.,  the  integral 

f'(t)    dt 


JL/lo! 

SvtJ  *»/(*) 


taken  round  the  circle,  is  zero  when  R  becomes  indefinitely  great.     But  the 
value  of  the  integral  is,  by  the  Corollary  in  §  20, 


'  (t)    6A 

+ 


!_   f<*>  J./'_(0  Jfc_        _L   y    ( 
27ri  J      V-  f(t)  t-z     2-n-i  <=1  J 


tn  f(fi    t-Z        2-7TI  J        tn  f(t)    t-Z        2lri  i=i  J         tn  f(t}  t-z' 

taken  round  small  circles  enclosing  the  origin,  the  point  z,  and  the  points 
a,i,  which  are  the  infinities  of  the  subject  of  integration;  the  origin  being 
supposed  a  zero  of /(t)  of  multiplicity  r. 

1     f»  !/'(*)    dt    ._!/'(*) 

JMOW 


tnf(t}t-Z       Znf(2}' 
dt  I         I 

»/ \^  /• 

Shr», 


1     fWlf(t 
iriJ      «"/(0 


L  f<0>  1£(Q  _^_         <^>(^)       r 
SwtJ     tnf(t)t-z          zn       zn+*' 

where  ^>  (^)  denotes  the  integral,  algebraical,  polynomial 


V  " f  +0  j~  i   -f        ~  if  +•••' 

when  t  is  made  zero.     Hence 

and  therefore 

which,  by  §  GO,  shews  that/(V)  is  of  class  n. 

*   Comptcs  Rendus,  t.  xciv,  (1882),  p.  G36. 


92  CLASS-PROPERTIES   OF  [61. 

COROLLARY.  The  product  of  any  finite  number  of  functions  of  the  same 
class  n  is  a  function  of  class  not  higher  than  n  ;  and  the  class  of  the  product 
of  any  finite  number  of  functions  of  different  classes  is  not  greater  than  the 
highest  class  of  the  component  functions. 

The  following  are  the  chief  references  to  memoirs  discussing  the  class  of  functions  : 

Laguerrc,  Comptes  Rendus,  t.  xciv,  (1882),  pp.  160-163,  pp.  635—638,  ib.  t.  xcv,  (1882), 
pp.  828—831,  ib.  t.  xcviii,  (1884),  pp.  79—81  ; 

Poincare,  Bull,  des  Sciences  Math.,  t.  xi,  (1883),  pp.  136—144  ; 

Cesaro,  Comptes  Rendm,  t.  xcix,  (1884),  pp.    26—27,  followed   (p.  27)  by   a  note  by 
Hermite;  Giornale  di  Battaglini,  t.  xxii,  (1884),  pp.  191  —  200; 

Vivanti,  Giornale  di  Battaglini,  t.  xxii,  (1884),  pp.  243—261,  pp.  378—380,  ib.  t.  xxiii, 
(1885),  pp.  96—122,  ib.  t.  xxvi,  (1888),  pp.  303—314  ; 

Hermite,  Cours  d  la  faculte'  des  Sciences  (4me  ed.,  1891),  pp.  91  —  93. 


Ex.  1.     The  function 


2 
1=1 


where  the  quantities  c  are  constants,  n  is  a  finite  integer,  and  the  functions  J\  (z)  are 
algebraical  polynomials,  is  of  class  unity. 

Ex.  2.     If  a  simple  function  be  of  class  %,  its  derivative  is  also  of  class  n. 

Ex.  3.     Discuss  the  conditions  under  which  the  sum  of  two  functions,  each  of  class  n, 
is  also  of  class  n. 

Ex.  4.     Examine  the  following  test  for  the  class  of  a  function,  due  to  Poincare. 

Let  a  be  any  number,  no  matter  how  small  provided  its  argument  be  such  that  eaz 
vanishes  when  z  tends  towards  infinity.     Then  /  (z)  is  of  class  n,  if  the  limit  of 


vanish  with  indefinite  increase  of  z. 

A  possible  value  of  a  is   2   ciai~n~1,  where  C;  is  a  constant  of  modulus  unity. 

Ex.  5.     Verify  the  following  test  for  the  class  of  a  function,  due  to  de  Sparre*. 

Let  X  be  any  positive  non-infinitesimal  quantity  ;  then  the  function  /  (z)  is  of  class  n, 
if  the  limit,  for  m  =  oo  ,  of 

\amn~l{\am  +  i\-\am\} 
be  not  less  than  X.     Thus  sin  z  is  of  class  unity. 

Ex.  6.     Let  the  roots  of  0n  +  1  =  l  be  1,  a,  a2,  ......  ,  an;   and  let  f  (s)  be  a  function 

of  class  n.    Then  forming  the  product 

n/(a«4 

we  evidently  have  an  integral  function  of  zn  +  1;  let  it  be  denoted  by  F(zn  +  1).     The  roots  of 
*  Comptes  Rendus,  t.  cii,  (1886),  p.  741. 


61.]  TRANSCENDENTAL    INTEGRAL    FUNCTIONS  93 

F(zn+l)  =  Q  are  a^'for  i=l,  2, and  s  =  0,  1, ,  n\  and  therefore,  replacing  zn  +  1  by  z, 

the  roots  ofF(z)  =  0  are  a?*1  for  i=l,  2,  ....... 

Since/  (z)  is  of  class  n,  the  series 


converges  uniformly  and  unconditionally.  This  series  is  the  sum  of  the  first  powers  of  the 
reciprocals  of  the  roots  of  F(z}~  0;  hence,  according  to  the  definition  (p.  89),  F(z)  is  of 
class  zero. 

It  therefore  follows  that  from,  a  function  of  any  class  a  function  of  class  zero  with  a 
modified  variable  can  be  deduced.  Conversely,  by  appropriately  modifying  the  variable  of 
a  given  function  of  class  zero,  it  is  possible  to  deduce  functions  of  any  required  class. 

Ex.  7.     If  all  the  zeros  of  the  function 

=1  r  anr 
\ 
be  real,  then  all  the  zeros  of  its  derivative  are  also  real.     (Witting.) 


00         I      /  ~  \ 

U\(l--)e' 
«=*  ^\      «W 


CHAPTER  VI. 

FUNCTIONS  WITH  A  LIMITED  NUMBER  OF  ESSENTIAL  SINGULARITIES. 

62.  SOME  indications  regarding  the  character  of  a  function  at  an 
essential  singularity  have  already  been  given.  Thus,  though  the  function 

is  regular  in  the  vicinity  of  such  a  point  a,  it  may,  like  sn  -  at  the  origin, 

% 

have  a  zero  of  unlimited  multiplicity  or  an  infinity  of  unlimited  multiplicity 
at  the  point ;  and  in  either  case  the  point  is  such  that  there  is  no  factor  of 
the  form  (z  —  a)x  which  can  be  associated  with  the  function  so  as  to  make  the 
point  an  ordinary  point  for  the  modified  function.  Moreover,  even  when 
the  path  of  approach  to  the  essential  singularity  is  specified,  the  value 
acquired  is  not  definite :  thus,  as  z  approaches  the  origin  along  the  axis  of  x, 

so  that  its  value  may  be  taken  to  be  1  -f-  (4>mK  +  x),  the  value  of  sn  -  is  not 

z 

definite  in  the  limit  when  m  is  made  infinite.  One  characteristic  of  the 
point  is  the  indefiniteness  of  value  of  the  function  there,  though  in  the 
vicinity  the  function  is  uniform. 

A  brief  statement  and  a  proof  of  this  characteristic  were  given  in  §  33 ; 
the  theorem  there  proved — that  a  uniform  analytical  function  can  assume 
any  value  at  an  essential  singularity — may  also  be  proved  as  follows.  The 
essential  singularity  will  be  taken  at  infinity — a  supposition  that  will  be 
found  not  to  detract  from  generality. 

Let  f(z)  be  a  function  having  any  number  of  zeros  and  any  number 
of  accidental  singularities  and  £  =  oo  for  its  sole  essential  singularity ;  then 
it  can  be  expressed  in  the  form 

/w-88*"' 

where  G1  (z)  is  algebraical  or  transcendental  according  as  the  number  of  zeros 
is  finite  or  infinite  and  G2(z)  is  algebraical  or  transcendental  according  as 
the  number  of  accidental  singularities  is  finite  or  infinite. 

If  Cr2  (z)  be  transcendental,  we  can  omit  the  generalising  factor  e°(z). 
Then  f(z)  has  an  infinite  number  of  accidental  singularities ;  each  of  them 
in  the  finite  part  of  the  plane  is  of  only  finite  multiplicity  and  therefore  some 
of  them  must  be  at  infinity.  At  each  such  point,  the  function  G2  (z)  vanishes 
and  Ol  (z)  does  not  vanish  ;  and  so  f(z)  has  infinite  values  for  z  =  oo  . 


62.]  VALUE   AT   AN   ESSENTIAL   SINGULARITY  95 

If  Gz  (2)  be  algebraical  and  Gl  (z)  be  also  algebraical,  then  the  factor  ea(z) 
may  not  be  omitted,  for  its  omission  would  make  f(z)  an  algebraical  function. 
Now  z  =  oo  is  either  an  ordinary  point  or  an  accidental  singularity  of 

ft  <*)/<?.<*); 

hence  as  g  (z}  is  integral  there  are  infinite  values  of  z  which  make 


infinite. 

If  G.>.(z)  be  algebraical  and  G^  (z)  be  transcendental,  the  factor  eg(z)  maybe 
omitted.     Let  al5  a2,...,  an  be  the  roots  of  G2(z):  then  taking 

f(z)=    ^- 


we  have  Ar= 

a  non-vanishing  constant  ;  and  so 


where  Gn  (z)  is  a  transcendental  integral  function.     When  2  =  oo  ,  the  value 
of  G3(z)/G.,(z)  is  zero,  but  Gn(z)  is  infinite  ;  hence  f(z)  has  infinite  values  for 

Z=  00  . 

Similarly  it  may  be  shewn,  as  follows,  that/(z)  has  zero  values  for  0  =  oo  . 

In  the  first  of  the  preceding  cases,  if  Gl  (z)  be  transcendental,  so  that  f  (z) 
has  an  infinite  number  of  zeros,  then  some  of  them  must  be  at  an  infinite 
distance;  f(z)  has  a  zero  value  for  each  such  point.  And  if  GI(Z)  be 
algebraical,  then  there  are  infinite  values  of  z  which,  not  being  zeros  of 
G2(z),  make  f(z)  vanish. 

In  the  second  case,  when  z  is  made  infinite  with  such  an  argument  as  to 
make  the  highest  term  in  g(z)  a  real  negative  quantity,  then  f(z)  vanishes 
for  that  infinite  value  of  z. 

In  the  third  case,/(V)  vanishes  for  a  zero  of  G1(z)  that  is  at  infinity. 

Hence  the  value  of  f  (z)  for  z=  oo  is  not  definite.  If,  moreover,  there 
be  any  value  neither  zero  nor  infinity,  say  G,  which  f(z)  cannot  acquire 
for  z  =  oo  ,  then 

/(*)-C 

is  a  function  which  cannot  be  zero  at  infinity  and  therefore  all  its  zeros  are 
in  the  finite  part  of  the  plane  :  no  one  of  them  is  an  essential  singularity,  for 
f(z)  has  only  a  single  value  at  any  point  in  the  finite  part  of  the  plane;  hence 
they  are  finite  in  number  and  are  isolated  points.  Let  H1  (z)  be  the  alge 
braical  polynomial  having  them  for  its  zeros.  The  accidental  singularities 
of  f(z}  —  C  are  the  accidental  singularities  of  f(z)  ;  hence 


96  FORM   OF   A   FUNCTION   NEAR  [62. 

where,  if  G2(z)  be  algebraical,  the  exponential  h(z)  must  occur,  since  f(z), 
and  therefore  f(z)  —  C,  is  transcendental.     The  function 

-|  sy     /     \ 

TJ1  ( ~\  —  _ 2  \   /  0—h  (z) 

•*  \*/      f  t  ~\      n      TT  f n\ 

J(z)-L      H1  (z) 

evidently  has  z=  oo  for  an  essential  singularity,  so  that,  by  the  second  or 
the  third  case  above,  it  certainly  has  an  infinite  value  for  z  =  co ,  that  is, 
f(z)  certainly  acquires  the  value  G  for  z=  GO  . 

Hence  the  function  can  acquire  any  value  at  an  essential  singularity. 

63.  We  now  proceed  to  obtain  the  character  of  the  expression  of  a 
function  at  a  point  z  which,  lying  in  the  region  of  continuity,  is  in  the 
vicinity  of  an  essential  singularity  b  in  the  finite  part  of  the  plane. 

With  b  as  centre  describe  two  circles,  so  that  their  circumferences  and 
the  whole  area  between  them  lie  entirely  within  the  region  of  continuity. 
The  radius  of  the  inner  circle  is  to  be  as  small  as  possible  consistent  with 
this  condition;  and  therefore,  as  it  will  be  assumed  that  b  is  the  only 
singularity  in  its  own  immediate  vicinity,  this  radius  may  be  made  very 
small. 

The  ordinary  point  z  of  the  function  may  be  taken  as  lying  within  the 
circular  ring-formed  part  of  the  region  of  continuity.  At  all  such  points  in 
this  band,  the  function  is  holomorphic  ;  and  therefore,  by  Laurent's  Theorem 
(§  28),  it  can  be  expanded  in  a  converging  series  of  positive  and  negative 
integral  powers  of  z  —  b  in  the  form 

+  V-L(Z  —  6)"1  +  v2  (z  —  6)~2  +  . . ., 
where  the  coefficients  un  are  determined  by  the  equation 

un  = 


the  integrals  being  taken  positively  round  the  outer  circle,  and  the  coefficients 
vn  are  determined  by  the  equation 


the  integrals  being  taken  positively  round  the  inner  circle. 

The  series  of  positive  powers  converges  everywhere  within  the  outer  circle 
of  centre  b,  and  so  (§  26)  it  may  be  denoted  by  P  (z  -  b)  ;  and  the  function  P 
may  be  either  algebraical  or  transcendental. 

The  series  of  negative  powers  converges  everywhere  without  the  inner 
circle  of  centre  b  ;  and,  since  6  is  not  an  accidental  but  an  essential  singularity 
of  the  function,  the  series  of  negative  powers  contains  an  infinite  number  of 


63.]  AN    ESSENTIAL   SINGULARITY  97 

terms.      It  may  be  denoted  by  G  I  --  rh  a  series  converging  for  all  points 

\z  —  o/ 

in  the  plane  except  z  =  b  and  vanishing  when  z  —  b  =  co. 


Thus 


is  the  analytical  representation  of  the  function  in  the  vicinity  of  its  essential 
singularity  b  ;  the  function  G  is  transcendental  and  converges  everywhere  in 
tlie  plane  except  at  z  =•  b,  and  the  function  P,  if  transcendental,  converges 
uniformly  and  unconditionally  for  sufficiently  small  values  of  |  z  —  b  \  . 

Had  the  singularity  at  b  been  accidental,  the  function  G  would  have  been 
algebraical. 

COROLLARY  I.  If  the  function  have  any  essential  singularity  other  than 
b,  it  is  an  essential  singularity  of  P  (z  —  b)  continued  outside  the  outer  circle  ; 

but  it  is  not  an  essential  singularity  of  G  (  --  j]  ,  for  the  latter  function 

\z  —  ol 

converges  everywhere  in  the  plane  outside  the  inner  circle. 

COROLLARY  II.  Suppose  the  function  has  no  singularity  in  the  plane 
except  at  the  point  b  ;  then  the  outer  circle  can  have  its  radius  made  infinite. 
In  that  case,  all  positive  powers  except  the  constant  term  w0  disappear: 
and  even  this  term  survives  only  in  case  the  function  have  a  finite  value  at 
infinity.  The  expression  for  the  function  is 


and  the  transcendental  series  converges  everywhere  outside  the  infinitesimal 
circle  round  b,  that  is,  everywhere  in  the  plane  except  at  the  point  b.  Hence 
the  function  can  be  represented  by 


This  special  result  is  deduced  by  Weierstrass  from  the  earlier  investiga 
tions*,  as  follows.  If  f(z)  be  such  a  function  with  an  essential  singularity  at 
b,  and  if  we  change  the  independent  variable  by  the  relation 

Z/==^b' 

thcn/(V)  changes  into  a  function  of  z',  the  only  essential  singularity  of  which 
is  at  /  =  GO  .  It  has  no  other  singularity  in  the  plane  ;  and  the  form  of  the 
function  is  therefore  G  (z'),  that  is,  a  function  having  an  essential  singularity 
at  b  but  no  other  singularity  in  the  plane  is 


*  Weierstrass  (I.e.),  p.  27. 
F. 


98  FORM   OF   A   FUNCTION   NEAR  [63. 

COROLLARY  III.  The  most  general  expression  of  a  function  having  its 
sole  essential  singularity  at  b  a  point  in  the  finite  part  of  the  plane  and  any 
number  of  accidental  singularities  is 


G, 

where  the  zeros  of  the  function  are  the  zeros  of  GI,  the  accidental  singularities 
of  the  function  are  the  zeros  of  G2,  and  the  function  g  in  the  exponential  is  a 
function  which  is  finite  everyiuhere  except  at  b. 

This  can  be  derived  in  the  same  way  as  before ;  or  it  can  be  deduced 
from  the  corresponding  theorem  relating  to  transcendental  integral  functions, 
as  above.  It  would  be  necessary  to  construct  an  integral  function  G2(z') 
having  as  its  zeros 


and   then   to  replace   z    by  -  — j  ;  and   G.,  is  algebraical  or  transcendental, 

Z  0 

according  as  the  number  of  zeros  is  finite  or  infinite. 
Similarly  we  obtain  the  following  result : 

COROLLARY  IV.  A  uniform  function  of  z,  which  has  its  sole  essential 
singularity  at  b  a  point  in  the  finite  part  of  the  plane  and  no  accidental 
singularities,  can  be  represented  in  the  form  of  an  infinite  product  of  primary 
factors  of  the  form 


\z  —  b 

which  converges  uniformly  and  unconditionally  everywhere  in  the  plane  except 
at  z  =  b. 

The   function   g  ( =]   is  an  integral  function  of T   vanishing  when 

J  \z-bj  z-b 

r  vanishes;  and  k  and  I  are  constants.     In  particular  factors,  q( T) 

z  -  b  ^  \z  -  b) 

may  vanish ;  and  either  k  or  I  (but  not  both  k  and  I)  may  vanish  with  or 

without  a  vanishing  exponent  q  { T  ) . 

J  \z-bj 

If  tt{  be  any  zero,  the   corresponding  primary  factor  may  evidently  be 
expressed  in  the  form 

(z  — 


,z  — 

Similarly,  for  a  uniform  function  of  z  with  its  sole  essential  singularity  at  b  and 
any  number  of  accidental  singularities,  the  product-form  is  at  once  derivable 


63.]  AN    ESSENTIAL   SINGULARITY  99 

by   applying   the   result   of  the   present    Corollary  to  the   result  given  in 
Corollary  III. 

These  results,  combined  with  the  results  of  Chapter  V.,  give  the  complete 
general  theory  of  uniform  functions  with  only  one  essential  singularity. 

64.  We  now  proceed  to  the  consideration  of  functions,  which  have  a 
limited  number  of  assigned  essential  singularities. 

The  theorem  of  §  63  gives  an  expression  for  the  function  at  any  point  in 
the  band  between  the  two  circles  there  drawn. 

Let  c  be  such  a  point,  which  is  thus  an  ordinary  point  for  the  function ; 
then  in  the  domain  of  c,  the  function  is  expansible  in  a  form  Pl  (z  —  c). 
This  domain  may  extend  to  an  essential  singularity  b,  or  it  may  be  limited 
by  a  pole  d  which  is  nearer  to  c  than  b  is,  or  it  may  be  limited  by  an 
essential  singularity  /  which  is  nearer  to  c  than  b  is.  In  the  first  case,  we 
form  a  continuation  of  the  function  in  a  direction  away  from  b;  in  the 
second  case,  we  continue  the  function  by  associating  with  the  function 
a  factor  (z  —  d)n  which  takes  account  of  the  accidental  singularity ;  in 
the  third  case,  we  form  a  continuation  of  the  function  towards  f.  Taking 
the  continuations  for  successive  domains  of  points  in  the  vicinity  of/,  we  can 
obtain  the  value  of  the  function  for  points  on  two  circles  that  have  /  for 
their  common  centre.  Using  these  values,  as  in  §  63,  to  obtain  coefficients, 
we  ultimately  construct  a  series  of  positive  and  negative  powers  converging 
except  at  /  for  the  vicinity  of/  Different  expressions  in  different  parts 
of  the  plane  will  thus  be  obtained,  each  being  valid  only  in  a  particular 
portion:  the  aggregate  of  all  of  them  is  the  analytical  expression  of  the 
function  for  the  whole  of  the  region  of  the  plane  where  the  function  exists. 

We  thus  have  one  mode  of  representation  of  the  function ;  its  chief 
advantage  is  that  it  indicates  the  form  in  the  vicinity  of  any  point,  though  it 
gives  no  suggestion  of  the  possible  modification  of  character  elsewhere.  This 
deficiency  renders  the  representation  insufficiently  precise  and  complete ;  and 
it  is  therefore  necessary  to  have  another  mode  of  representation. 

65.  Suppose  that  the  function  has  n  essential  singularities  a,!,  a*,...,  an 
and  that  it   has  no  other  singularity.     Let  a  circle,  or  any  simple  closed 
curve,  be  drawn  enclosing  them  all,  every  point  of  the  boundary  as  well 
as  the  included   area  (with  the  exception  of  the  n  singularities)  lying  in 
the  region  of  continuity  of  the  function. 

Let  z  be  any  ordinary  point  in  the  interior  of  the  circle  or  curve ;  and 
consider  the  integral  ,  f/+\ 

I '*=-***> 

taken  round  the  curve.     If  we  surround  z  and  each  of  the  n  singularities  by 
small  circles  with  the  respective  points  for  centres,  then  the  integral  round 

7—2 


100  FUNCTIONS   WITH    A    LIMITED   NUMBER  [65. 

the  outer  curve  is  equal  to  the  sum  of  the  values  of  the  integral  taken  round 
the  n  +  l  circles.     Thus 


and  therefore 


The  left-hand  side  of  the  equation  isf(z). 
Evaluating  the  integrals,  we  have 


where  Gr  is,  as  before,  a  transcendental  function  of  -  - —  vanishing  when 

1 

is  zero. 

z  —  ar 

Now,  of  these  functions,  Gr{-     -]  converges  everywhere  in  the  plane 

\&        \jbfj 

except  at  ar :  and  therefore,  as  n  is  finite, 


r=i      \z  -  a 
is  a  function  which  converges  everywhere  in  the  plane  except  at  the  n  points 

Clj , . . . ,  an . 

Because  z  =  oc  is  not  an  essential  singularity  of  f(z),  the  radius  of  the 

circle  in  the  integral  =—. .  !  /--  dt  may  be  indefinitely  increased.    The  value 

ZTTI  J  s  t  —  z 

of  f(t)  tends,  with  unlimited  increase  of  t,  to  some  determinate  value  G  which 
is  not  infinite  ;  hence,  as  in  §  24,  II.,  Corollary,  the  value  of  the  integral  is 
C.  We  therefore  have  the  result  that/0)  can  be  expressed  in  the  form 


\z-a, 
or,  absorbing  the  constant  C  into  the  functions  G  and  replacing  the  limitation, 

that  the  function  Gr(— — }   shall  vanish  for  —  =  0,  by  the  limitation 

\z  —  arj  z  —  ar 

that,  for  the  same  value  =0,  it  shall  be  finite,  we  have  the  theorem*:— 

z  —  ar 

If  a  given  function  f(z)  have  n  singularities  a^,...,  an,  all  of  which  are  in 
the  finite  part  of  the  plane  and  are  essential  singularities,  it  can  be  expressed 
in  the  form 

2G  f-M, 

r=i   r  \z  -  aj  ' 

*  The  method  of  proof,  by  an  integration,  is  used  for  brevity  :  the  theorem  can  be  established 
by  purely  algebraical  reasoning. 


65.]  OF   ESSENTIAL   SINGULARITIES  101 

where  Gr  is  a  transcendental  function  converging  everywhere  in  the  plane 

except  at  ar  and  having  a  determinate  finite  value  gr  for  -     —  =  0,  such 

z  —  cir 

n 

that  2  gr  is  the  finite  value  of  the  given  function  at  infinity. 
r=l 

COROLLARY.  If  the  given  function  have  a  singularity  at  oo  ,  and  n  singu 
larities  in  the  finite  part  of  the  plane,  then  the  function  can  be  expressed  in 
the  form 


w        /     1     \ 
G(z)  +  SG,(— L-J, 

r=i      \z-arr 


where  Gr  is  a  transcendental  or  an  algebraic  polynomial  function,  according 
as  ar  is  an  essential  or  an  accidental  singularity :  and  so  also  for  G  (z),  accord 
ing  to  the  character  of  the  singularity  at  infinity. 

66.     Any  uniform  function,  which  has  an  essential  singularity  at  z  =  a, 
can  (§  63)  be  expressed  in  the  form 


for  points  z  in  the  vicinity  of  a.  Suppose  that,  for  points  in  this  vicinity, 
the  function  f(z)  has  no  zero ;  that  it  has  no  accidental  singularity ;  and 
therefore,  among  such  points  z,  the  function 

1    df(z) 
/(*)    dz 

has  no  pole,  and  therefore  no  singularity  except  that  at  a  which  is  essential. 
Hence  it  can  be  expanded  in  the  form 

G(^+P(z-a\ 


z-a 


where  G  converges  everywhere  in  the  plane  except  at  a,  and  vanishes  for 
=  0.     Let 


z  —  a     dz 


,  /    1    \ 

where  0^  I  ^— ^  I  converges  everywhere  in  the  plane  except  at  a,  and  vanishes 

for  — —  =  o. 

z  —  a 

Then  c,  evidently  not  an  infinite  quantity,  is  an  integer.     To  prove  this, 
describe  a  small  circle  of  radius  p  round  a :  then  taking  z-a  =  pe91  so  that 

—  =  idd,  we  have 
z  —  a 

l     M(*\ 

dz  =  P  (z  —  a)  dz 


102  FUNCTIONS   WITH   A   LIMITED   NUMBER  [66. 

and  therefore 


Now  JP(z  —  a)dz  is  a  uniform  function  :  and  so  is  f(z).  But  a  change 
of  6  into  6  +  2-7T  does  not  alter  z  or  any  of  the  functions  :  thus 

actotr  —  1   • 

~~  *•  i 

and  therefore  c  is  an  integer. 

67,  If  the  function  /(z)  have  essential  singularities  alt...,  an  and  no 
others,  then'  it  can  be  expressed  in  the  form 

n          /I 

C+  $9J-± 

r=i     \z-ar 

If  there  be  no  zeros  for  this  function  f(z)  anywhere  (except  of  course  such 
as  may  enter  through  the  indeterminateness  at  the  essential  singularities), 
then 


/(*)     dz 

has  n  essential  singularities  a1}...,  an  and  no  other  singularities  of  any  kind. 
Hence  it  can  be  expressed  in  the  form 


n         /     1     \ 
C+  2  Gr(-    -), 
r=i       \z-a,rl 


where  the  function  Gr  vanishes  with .     Let 

z  —  ar 

cr          d 

I         T~~ 


\js  —  a,./      z  —  ar     dz  {   r\z  —  ar 

where  Gr  I )  is  a  function  of  the  same  kind  as  Gr  ( ) . 

\z  —  ar/  \z  —  arj 

Then  all  the  coefficients  cr,  evidently  not  infinite  quantities,  are  integers. 
For,  let  a  small  circle  of  radius  p  be  drawn  round  ar :  then,  if  z  —  ar  =  peei,  we 
have 

crdz 


z  —  ar 


=  cri6, 


and  — ^ —  =  dPs  (z  -  ar). 

z  —  as 

We  proceed  as  before :  the  expression  for  the  function  in  the  former 
case  is  changed  so  that  now  the  sum  2Pg(0—  ar)  for  5  =  !,...,  i — 1, 
r  +  1,...,  n  is  a  uniform  function;  there  is  no  other  change.  In  exactly  the 
same  way  as  before,  we  shew  that  every  one  of  the  coefficients  cr  is  an 
integer. 

Hence  it  appears  that  if  a  given  function  f(z)  have,  in  the  finite  part  of 


67.]  OF   ESSENTIAL   SINGULARITIES  103 

the  plane,  n  essential  singularities  al,...,  an  and  no  other  singularities  and  if 
it  have  no  zeros  anywhere  in  the  plane,  then 


f(z)     dz 

where  all  the  coefficients  c*  are  integers,  and  the  functions  G  converge  every 
where  in  the  plane  except  at  the  essential  singularities  and  Gi  vanishes  for 

-J--  0. 


Now,  since  f(z)  has  no  singularity  at  oo  ,  we  have  for  very  large  values  of  z 


and  /'W  =  _>_ 

Z* 

and  therefore,  for  very  large  values  of  z, 


_ 

f(z)    dz  u0  z2      z3 

Thus  there  is  no  constant  term  in  =7-^   ^r-^  ,  and  there  is  no  term  in  -.   But 

/(*)     dz  z 

the  above  expression  for  it  gives  G  as  the  constant  term,  which  must  therefore 

vanish  ;  and  it  gives  2c;  as  the  coefficient  of  -  ,  for  -7-  •<  (r<  (  -     —  H  will  begin 

z          dz  [      \z  —  ftj/  J 

with  —  at  least  ;  thus  ^a  must  therefore  also  vanish. 

Z" 

Hence  for  a  function  f  (z)  which  has  no  singularity  at  z=  oo  and  no 
zeros  anywhere  in  the  plane  and  of  which  the  only  singularities  are  the  n 
essential  singularities  at  a1}  a2,...,  an,  we  have 


/  (z)    dz        i=i  z  -  Oi      i=i  dz  (      \z-  a 
where  the  coefficients  a  are  integers  subject  to  the  condition 

n 

2  ct  =  0. 

i=l 

If  an=  oo  ,  so  that  2=  GO  is  an  essential  singularity  in  addition  to  a2,  a2,..., 
an_j,  there  is  a  term  6r  (z)  instead  of  Gn(  —     -  ]  ;  there  is  no  term,  that  corre- 

\Z  —  C^n/ 
/^ 

spends  to  -     —  ,  but  there  may  be  a  constant  G.     Writing 

— 


z  — 


with  the  condition  that  G  (z)  vanishes  when  z  —  0,  we  then  have 


- 
=        __  g  ^ 

i=iz-at     dz(    v  /J      ». 


104  PRODUCT-EXPRESSION    OF  [67. 

where    the   coefficients   d  are   integers,  but   are  no   longer  subject  to  the 
condition  that  their  sum  vanishes. 

Let  R*  (z}  denote  the  function 


the  product  extending  over  the  factors  associated  with  the  essential  sin 
gularities  of  f(z)  that  lie  in  the  finite  part  of  the  plane;  thus  R*(z)  is  a 
rational  algebraical  rneromorphic  function.  Since 

1      dR*(z)  =  2      d 
R*  (z)     dz     ~  i=\z  —  a,i' 
we  have 

1    df(z)  _      1      dR*(z)  =$    d_(-Q  (    * 
f(z)    dz        R* (z)      dz         i=\dz\   l\z  —  a^ 

where  Gn  ( —     -  )  is  to  be  replaced  by  G  (z)  if  an  =  <x> ,  that  is,  if  z  —  oo  be  an 
\z  —  anj 

essential  singularity  off(z).  Hence,  except  as  to  an  undetermined  constant 
factor,  we  have 


t=i 

which  is  therefore  an  analytical  representation  of  a  function  with  n  essential 
singularities,  no  accidental  singularities,  and  no  zeros:  and  the  rational  alge 
braical  function  R*  (z)  becomes  zero  or  oo  only  at  the  singularities  off(z). 

If  z  =  oo  be  not  an  essential  singularity,  then  R*  (z)  for  z  =  oo  is  equal  to 

M 

unity  because  2  Cf  =  0. 
1=1 

COROLLARY.  It  is  easy  to  see,  from  §  43,  that,  if  the  point  a;  be  only  an 
accidental  singularity,  then  a  is  a  negative  integer  and  wj  I  —  -  )  is  zero :  so 

\Z  —  Oii/ 

that  the  polar  property  at  c^  is  determined  by  the  occurrence  of  a  factor 
(z  —  a{)Ci  solely  in  the  denominator  of  the  rational  meromorphic  function  R*  (z). 

And,  in  general,  each  of  the  integral  coefficients  a  is  determined  from  the 
expansion  of  the  function  f'(z)  +f(z)  in  the  vicinity  of  the  singularity 
with  which  it  is  associated. 

68.  Another  form  of  expression  for  the  function  can  be  obtained  from 
the  preceding;  and  it  is  valid  even  when  the  function  has  zeros  not 
absorbed  into  the  essential  singularities  f. 

Consider  a  function  with  one  essential  singularity,  and  let  a  be  the 
point ;  and  suppose  that,  within  a  finite  circle  of  centre  a  (or  within  a  finite 
simple  curve  which  encloses  a),  there  are  m  simple  zeros  a,  /3,...,  X  of  the 

+  See  Guichard,  TMorie  des  points  singuliers  essentiels,  (These,  Gauthier-Villars,  Paris,  1883), 
especially  the  first  part. 


68.]  A    FUNCTION  105 

function  f(z)  —  m  being  assumed  to  be  finite,  and  it  being  also  assumed  that 
there  are  no  accidental  singularities  within  the  circle.     Then,  if 

/(*)  =  (*  -  «)  (z  -  /3).  .  .(*  -\)F  (z\ 

the  function  F  (z)  has  a  for  an  essential  singularity  and  has  no  zeros  within 
the  circle.     Hence,  for  points  z  within  the  circle, 


where  (?,  (  -----  )  converges  everywhere  in  the  plane  and  vanishes  with  -     —  , 

\z  —  a]  z     a 

and  P(z  —  a)  is  an  integral  function  converging  uniformly  and  unconditionally 
within  the  circle  ;  moreover,  c  is  an  integer.     Thus 


F(z)  =  A(z-  a)"  eGl 
Let       (*-a)(*-/3)...(*-X)  =  (*-a 

_  (  y  _  r,  \m 

a) 


then  f(z)  =  (*  -  dTffl    --    F(z} 

\z      u/ 

=  A(z-  ar+°gi  (~}eG>  ^  e  ^~a]  "z  . 
\z  —  ft/ 

Now  of  this  product-  expression  for/(V)  it  should  be  noted:  — 

(i)     That  m  +  c  is  an  integer,  finite  because  m  and  c  are  finite  : 


<?,—- 
(ii)     The  function  e  '  ^z~a'  can  be  expressed  in  the  form  of  a  series  con 

verging    uniformly   and    unconditionally    everywhere,   except    at    z  =  a,   and 

proceeding  in  powers  of  -     —  in  the  form 

z     a 


.... 

z  —  a     (z  —  af 

It  has  no  zero  within  the  circle  considered,  for  F  (z)  has  no  zero.     Also  gl(-    -  1 

\z     a/ 

algebraical  function  of  —  '  —  ,  beginning  with  unity  and  containing  only 


is  an 

z  —  a 


a  finite  number  of  terms  :  hence,  multiplying  the  two  series  together,  we  have 

as  the  product  a  series  proceeding  in  powers  of  -  in  the  form 

£  ~"  a 


—  a 


which  converges  uniformly  and  unconditionally  everywhere  outside  any  small 
circle  round  a,  that  is,  everywhere  except  at  a.     Let  this  series  be  denoted  by 


106  PRODUCT-EXPRESSION    OF  [68. 

H  I ];  it  has  an   essential  singularity  at  a  and  its  only  zeros  are  the 

\z-aj 

points  a,  (3,...,  X,  for  the  series  multiplied  by  gl  (—    -)  has 

\z  —  ft/ 


no  zeros : 


(iii)  The  function  fP  (z  —  a)  dz  is  a  series  of  positive  powers  of  z  —  a, 
converging  uniformly  in  the  vicinity  of  a;  and  therefore  Q^(z-d)dz  can  ke 
expanded  in  a  series  of  positive  integral  powers  of  z  —  a  which  converges 
in  the  vicinity  of  a.  Let  it  be  denoted  by  Q  (z  —  a)  which,  since  it  is  a 
factor  of  F  (z),  has  no  zeros  within  the  circle. 

Hence  we  have 


/(*)  =  A  (z  -  aYQ  (z  -  a)  H 
where  p,  is  an  integer  ;  H  (  —    -  J  is  a  series  that  converges  everywhere  except 

at  a,  is  equal  to  unity  when  -         vanishes,  and  has  as  its  zeros  the  (finite) 

z  —  a 

number  of  zeros  assigned  to  f(z)  within  a  finite  circle  of  centre  a  ;  and 
Q  (z  —  a)  is  a  series  of  positive  powers  of  z  —  a  beginning  with  unity  which 
converges  —  but  has  no  zero  —  within  the  circle. 

The  foregoing  function  f(z)  is  supposed  to  have  no  essential  singularity 
except  at  ft.  If,  however,  a  given  function  have  singularities  at  points 
other  than  a,  then  the  circle  would  be  taken  of  radius  less  than  the  distance 
of  a  from  the  nearest  essential  singularity. 

Introducing  a  new  function  f{  (z}  defined  by  the  equation 


the  value  of  /[  (z)  is  Q  (z  —  a)  within  the  circle,  but  it  is  not  determined  by 
the  foregoing  analysis  for  points  without  the  circle.  Moreover,  as  (z  —  a)* 

and  also  Hi—       ]   are  finite  everywhere  except  possibly  at   a,  it   follows 

that  essential  singularities  of  f(z)  other  than  a  must  be  essential  singu 
larities  of  fj  (z).  Also  since  /i  (z)  is  Q(z  —  a)  in  the  immediate  vicinity  of  a, 
this  point  is  not  an  essential  singularity  of  /i  (z). 

Thus  /i  (z)  is  a  function  of  the  same  kind  as  f(z)  ;  it  has  all  the  essential 
singularities  of  f(z)  except  ft,  but  it  has  fewer  zeros,  on  account  of  the  m 

zeros  of  f(z)  possessed  by  H  (  -    —  ]  .      The  foregoing  expression  for  f(z)  is 

\Z  —  ft/ 

the  one  referred  to  at  the  beginning  of  the  section. 

If  we  choose  to  absorb  into  /x  (z}  the  factors  e     \z~a'   and   e?P(z~°  dz, 
which  occur  in 


(z  -  $*•  ffl  f  Jil  ^  (T-  a) 

\2  —  ft/ 


68.]  A    FUNCTION  107 

an  expression  that  is  valid  within  the  circle  considered,  then  we  obtain  a 
result  that  is  otherwise  obvious,  by  taking 


where  now  g±  (—    —  )  is  algebraical  and  has  for  its  zeros  all  the  zeros  within 

\Z  —  d/ 

the  circle  ;  yu,  is  an  integer;  and/j  (z)  is  a  function  of  the  same  kind  as  f(z), 
which  now  possesses  all  the  essential  singularities  of  f(z}  but  has  zeros  fewer 

by  the  in  zeros  that  are  possessed  by 


z—  a 


69.  Next,  consider  a  function  f(z)  with  n  essential  singularities  al} 
a2,...,  an  but  without  accidental  singularities;  and  let  it  have  any  number  of 
zeros. 

When  the  zeros  are  limited  in  number,  they  may  be  taken  to  be  isolated 
points,  distinct  in  position  from  the  essential  singularities. 

When  the  zeros  are  unlimited  in  number,  then  at  least  one  of  the 
singularities  must  be  such  that  an  infinite  number  of  the  zeros  lie  within 
a  circle  of  finite  radius,  described  round  it  as  centre  and  containing  no  other 
singularity.  For  if  there  be  not  an  infinite  number  in  such  a  vicinity 
of  some  one  point  (which  can  only  be  an  essential  singularity,  otherwise  the 
function  would  be  zero  everywhere),  then  the  points  are  isolated  and  there 
must  be  an  infinite  number  at  z  =  oo  .  If  z  =  oo  be  an  essential  singularity,  the 
above  alternative  is  satisfied  :  if  not,  the  function,  being  continuous  save  at 
singularities,  must  be  zero  at  all  other  parts  of  the  plane.  Hence  it  follows 
that  if  a  uniform  function  have  a  finite  number  of  essential  singularities  and 
an  infinite  number  of  zeros,  all  but  a  finite  number  of  the  zeros  lie  within 
circles  of  finite  radii  described  round  the  essential  singularities  as  centres  ; 
at  least  one  of  the  circles  contains  an  infinite  number  of  the  zeros,  and  some 
of  the  circles  may  contain  only  a  finite  number  of  them. 

We  divide  the  whole  plane  into  regions,  each  containing  one  but  only  one 
singularity  and  containing  also  the  circle  round  the  singularity  ;  let  the 
region  containing  a{  be  denoted  by  Ci,  and  let  the  region  Gn  be  the  part  of 
the  plane  other  than  Glt  (72,  ......  ,  Gn_^. 

If  the  region  G1  contain  only  a  limited  number  of  the  zeros,  then,  by  §  68, 
we  can  choose  a  new  function  /i  (z)  such  that,  if 


the  function  /j  (z)  has  av  for  an  ordinary  point,  has  no  zeros  within  the  region 
Glt  and  has  a2,  a3,  ......  ,  an  for  its  essential  singularities. 

If  the  region  Cl  contain  an  unlimited  number  of  the  zeros,  then,  as  in 
Corollaries  II.  and  III.  of  §   63,   we   construct   any  transcendental  function 


108  GENERAL  FORM  OF  A  FUNCTION  [69. 


5xf—     —  )  ,  having  a^  for  its  sole  essential  singularity  and  the  zeros  in  GI  for 

\z  —  OiJ 

all  its  zeros.     When  we  introduce  a  function  g:  (2),  defined  by  the  equation 


the  function  g:{z)  has  no  zeros  in  GI  and  certainly  has  a2,  a3,  ......  ,  an  for 

essential  singularities  ;  in  the  absence  of  the  generalising  factor  of  Glt  it  can 
have  Hi  for  an  essential  singularity.     By  §  G7,  the  function  ~g{  (2),  defined  by 

gi  (z)  =  0  -  cOc'  ehl  ^W  , 

has  no  zero  and  no  accidental  singularity,  and  it  has  a^  as  its  sole  essential 
singularity  :  hence,  properly  choosing  cx  and  hi,  we  may  take 

ft(*)-?i(*)/i(*)« 

so  that  fi  (z)  does  not  have  aj  as  an  essential  singularity,  but  it  has  all  the 
remaining  singularities  of  ^  (z),  and  it  has  no  zeros  within  C^. 
In  either  case,  we  have  a  new  function  ft  (z)  given  by 


where  /^  is  an  integer,  the  zeros  off(z)  that  lie  in  GI  are  the  zeros  of  GI  ;  the 
function  fi(z)  has  «2,  »s>  ......  >  an  (but  not  a^  for  its  essential  singularities, 

and  it  has  the  zeros  of  f(z)  in  the  remaining  regions  for  its  zeros. 
Similarly,  considering  (72,  we  obtain  a  function  /2  (z),  such  that 


where  /A.2  is  an  integer,  G2  is  a  transcendental  function  finite  everywhere  except 
at  a2  and  has  for  its  zeros  all  the  zeros  of  ft  (z)  —  and  therefore  all  the  zeros  of 
f  (z)  —  that  lie  in  G2  ;  then  f.2  (z)  possesses  all  the  zeros  of  f(z)  in  the  regions 
other  than  GI  and  C2,  and  has  a3,  a4,...,  an  for  its  essential  singularities. 

Proceeding  in  this  manner,  we  ultimately  obtain  a  function  fn  (z)  which 
has  none  of  the  zeros  off(z)  in  any  of  the  n  regions  GI,  C2,...,  Cn,  that  is,  has 
no  zeros  in  the  plane,  and  it  has  no  essential  singularities  ;  it  has  no  acci 

dental   singularities,  and  therefore  fn(z)  is  a  constant.      Hence,    when   we 

• 
substitute,  and  denote  by  S*  (z}  the  product  II  (z  —  a^1,  we  have 


Z  — 

as  the  most  general  form  of  a  function  having  n  essential  singularities,  no 
accidental  singularities,  and  any  number  of  zeros.  The  function  S*  (z)  is  a 
rational  algebraical  function  of  z,  usually  meromorphic  inform,  and  it  has  the 
essential  singularities  off(z)  as  its  zeros  and  poles ;  and  the  zeros  of  f  (z)  are 
distributed  among  the  functions  Gt. 

As  however  the  distribution  of  the  zeros  by  the  regions  C  and  therefore 


69.]  WITH    ESSENTIAL   SINGULARITIES  109 

the  functions  G[ )  are  somewhat  arbitrary,  the  above  form  though  general 

\z  —  a] 

is  not  unique. 

If  any  one  of  the  singularities,  say  am,  had  been  accidental  and  not 

essential,  then  in  the  corresponding  form  the  function  Gm  ( -       - )  would  be 

\Z  —  dm/ 

algebraical  arid  not  transcendental. 

70.  A  function  f(z],  which  has  any  finite  number  of  accidental  singu 
larities  in  addition  to  n  assigned  essential  singularities  and  any  number  of 
assigned  zeros,  can  be  constructed  as  follows. 

Let  A  (z)  be  the  algebraical  polynomial  which  has,  for  its  zeros,  the 
accidental  singularities  of  f(z),  each  in  its  proper  multiplicity.  Then  the 
product 

/(*)-A(*) 

is  a  function  which  has  no  accidental  singularities  ;  its  zeros  and  its  essential 
singularities  are  the  assigned  zeros  and  the  assigned  essential  singularities  of 
f  (z)  and  therefore  it  is  included  in  the  form 


n  ( 
S*(z)U  \0t 

i=i  ( 


where  S*  (z)  is  a  rational  algebraical  meromorphic  function  having  the  points 
Oi,  a.,,...,  an  for  zeros  and  poles.     The  form  of  the  function  f(z}  is  therefore 


A 


)}• 

-ail) 


71.  A  function  f  (z),  which  has  an  unlimited  number  of  accidental  singu 
larities  in  addition  to  n  assigned  essential  singularities  and  any  number  of 
assigned  zeros,  can  be  constructed  as  follows. 

Let  the  accidental  singularities  be  of, /3',....  Construct  a  function  f^ (z), 
having  the  n  essential  singularities  assigned  to  f  (z},  no  accidental  singu 
larities,  and  the  series  a!,  /3',. . .  of  zeros.  It  will,  by  §  69,  be  of  the  form  of  a 
product  of  n  transcendental  functions  Gn+1,...,  G.2n  which  are  such  that  a 
function  G  has  for  its  zeros  the  zeros  oif-i(z}  lying  within  a  region  of  the  plane, 
divided  as  in  §  69 ;  and  the  function  Gn+t  is  associated  with  the  point  at-. 

Thus  /  (z)  =  T*-(z)  ft  Gn 

f=i 

where  T*  (z)  is  a  rational  algebraical  meromorphic  function  having  its  zeros 
and  its  poles,  each  of  finite  multiplicity,  at  the  essential  singularities  ofy(^). 
Because  the  accidental  singularities  of  f(z)  are  the  same  points  and  have 
the  same  multiplicity  as  the  zeros  of  /i  (z),  the  function  /  (z)  /x  (z)  has  no 
accidental  singularities.  This  new  function  has  all  the  zeros  of  f(z),  and 
al,...,an  are  its  essential  singularities;  moreover,  it  has  no  accidental  singu 
larities.  Hence  the  product  f(z)fi  (z)  can  be  represented  in  the  form 


110  GENERAL  FORM  OF  A  FUNCTION  [71. 


and  therefore  we  have 


z-fi 


(f-a) 


as  an  expression  of  the  function. 

But,  as  by  their  distribution  through  the  n  selected  regions  of  the  plane 
in  §  69,  the  zeros  can  to  some  extent  be  arbitrarily  associated  with  the 
functions  Gl}  G*,,...,  Gn  and  likewise  the  accidental  singularities  can  to  some 
extent  be  arbitrarily  associated  with  the  functions  Gn+l,  Gn+»,...,  G^i,  the 
product-expression  just  obtained,  though  definite  in  character,  is  not  unique 
in  the  detailed  form  of  the  functions  which  occur. 

S*  (z) 
The  fraction  7**)  \ 

is  algebraical  and  rational ;  and  it  vanishes  or  becomes  infinite  only  at  the 
essential  singularities  alt  a.2,...,  an,  being  the  product  of  factors  of  the  form 
(z  —  «i)ms  for  i  =  l,  2,...,  n.  Let  the  power  (z  —  a^  be  absorbed  into  the 
function  G{/Gn+i  for  each  of  the  n  values  of  i ;  no  substantial  change  in  the 
transcendental  character  of  Gi  and  of  Gn+i  is  thereby  caused,  and  we  may 
therefore  use  the  same  symbol  to  denote  the  modified  function  after  the 
absorption.  Hence  "f"  the  most  general  product-expression  of  a  uniform 
function  of  z  which  has  n  essential  singularities  al}  a*,...,  an,  any  unlimited 
number  of  assigned  zeros  and  any  unlimited  number  of  assigned  accidental 
singularities  is 

n      ^ 

n  — 


\z-an 

The  resolution  of  a  transcendental  function  with  one  essential  singularity 
into  its  primary  factors,  each  of  which  gives  only  a  single  zero  of  the  function, 
has  been  obtained  in  §  63,  Corollary  IV. 

We  therefore  resolve  each  of  the  functions  G^...,  Gm  into  its  primary 
factors.  Each  factor  of  the  first  n  functions  will  contain  one  and  only  one  zero 
of  the  original  functions  / (.z) ;  and  each  factor  of  the  second  n  functions  will 
contain  one  and  only  one  of  the  poles  of  f(z).  The  sole  essential  singularity 
of  each  primary  factor  is  one  of  the  essential  singularities  off(z).  Hence  we 
have  a  method  of  constructing  a  uniform  function  with  any  finite  number  of 
essential  singularities  as  a  uniformly  converging  product  of  any  number  of 
primary  factors,  each  of  which  has  one  of  the  essential  singularities  as  its  sole 
essential  singularity  and  either  (i)  has  as  its  sole  zero  either  one  of  the  zeros 

t  Weierstrass,  I.e.,  p.  48. 


71.]  WITH    ESSENTIAL   SINGULARITIES  111 

or  one  of  the  accidental  singularities  of/(V),  so  that  it  is  of  the  form 


Z  —  €  \     a  (  — . 


or  (ii)  it  has  no  zero  and  then  it  is  of  the  form 

/fe). 

When  all  the  primary  factors  of  the  latter  form  are  combined,  they  constitute 
a  generalising  factor  in  exactly  the  same  way  as  in  §  52  and  in  §  63, 
Cor.  III.,  except  that  now  the  number  of  essential  singularities  is  not 
limited  to  unity. 

Two  forms  of  expression  of  a  function  with  a  limited  number  of  essential 
singularities  have  been  obtained :  one  (§  65)  as  a  sum,  the  other  (§  69)  as  a 
product,  of  functions  each  of  which  has  only  one  essential  singularity.  Inter 
mediate  expressions,  partly  product  and  partly  sum,  can  be  derived,  e.g. 
expressions  of  the  form 


z—  c. 


But  the  pure  product-expression  is  the  most  general,  in  that  it  brings  into 
evidence  not  merely  the  n  essential  singularities  but  also  the  zeros  and  the 
accidental  singularities,  whereas  the  expression  as  a  sum  tacitly  requires  that 
the  function  shall  have  no  singularities  other  than  the  n  which  are  essential. 

Note.  The  formation  of  the  various  elements,  the  aggregate  of  which  is  the  complete 
representation  of  the  function  with  a  limited  number  of  essential  singularities,  can  be 
carried  out  in  the  same  manner  as  in  §  34 ;  each  element  is  associated  with  a  particular 
domain,  the  range  of  the  domain  is  limited  by  the  nearest  singularities,  and  the  aggregate 
of  the  singularities  forms  the  boundary  of  the  region  of  continuity. 

To  avoid  the  practical  difficulty  of  the  gradual  formation  of  the  region  of  continuity 
by  the  construction  of  the  successive  domains  when  there  is  a  limited  number  of 
singularities  (and  also,  if  desirable  to  be  considered,  of  branch-points),  Fuchs  devised 
a  method  which  simplifies  the  process.  The  basis  of  the  method  is  an  appropriate  change 
of  the  independent  variable.  The  result  of  that  change  is  to  divide  the  plane  of  the 
modified  variable  f  into  two  portions,  one  of  which,  G2,  is  finite  in  area  and  the  other  of 
which,  Gl,  occupies  the  rest  of  the  plane;  and  the  boundary,  common  to  Gl  and  G2,  is  a 
circle  of  finite  radius,  called  the  discriminating  circle*  of  the  function.  In  G2  the 
modified  function  is  holomorphic ;  in  G^  the  function  is  holomorphic  except  at  f  =  oo  ; 
and  all  the  singularities  (and  the  branch-points,  if  any)  lie  on  the  discriminating  circle. 

The  theory  is  given  in  Fuchs's  memoir  "  Ueber  die  Darstellung  der  Functionen  com- 

plexer  Variabeln, ,"  Crelle,  t.  Ixxv,  (1872),  pp.  176 — 223.  It  is  corrected  in  details 

and  is  amplified  in  Crelle,  t.  cvi,  (1890),  pp.  1 — 4,  and  in  Crelle,  t.  cviii,  (1891), 
pp.  181—192;  see  also  Nekrassoff,  Math.  Ann.,  t.  xxxviii,  (1891),  pp.  82—90,  and 
Anissimoff,  Math.  Ann.,  t.  xl,  (1892),  pp.  145—148. 

*  Fuchs  calls  it  Grenzkreis. 


CHAPTER   VII. 

FUNCTIONS  WITH  UNLIMITED  ESSENTIAL  SINGULARITIES,  AND  EXPANSION 

IN  SERIES  OF  FUNCTIONS. 

72.  IT  now  remains  to  consider  functions  which  have  an  infinite  number 
of  essential  singularities*.  It  will,  in  the  first  place,  be  assumed  that  the 
essential  singularities  are  isolated  points,  that  is,  that  they  do  not  form  a 
continuous  line,  however  short,  and  that  they  do  not  constitute  a  continuous 
area,  however  small,  in  the  plane.  Since  their  number  is  unlimited  and 
their  distance  from  one  another  is  finite,  there  must  be  at  least  one  point  in 
the  plane  (it  may  be  at  z  =  oo  )  where  there  is  an  infinite  aggregate  of  such 
points.  But  no  special  note  need  be  taken  of  this  fact,  for  the  character  of  an 
essential  singularity  has  not  yet  entered  into  question ;  the  essential  singu 
larity  at  such  a  point  would  merely  be  of  a  nature  different  from  the  essential 
singularity  at  some  other  point. 

We  take,  therefore,  an  infinite  series  of  quantities  a1}  a.2,  a3,...  arranged  in 
order  of  increasing  moduli,  and  such  that  no  two  are  the  same :  and  so  we 
have  infinity  as  the  limit  of  av  when  v  =  <x> . 

Let  there  be  an  associated  series  of  uniform  functions  of  z  such  that 
for  all  values  of  i.  the  function  G'i  ( ) ,  vanishing  with ,  has  a{  as  its 

\Z  -  Of/  Z  —  Oi 

*  The  results  in  the  present  chapter  are  founded,  except  where  other  particular  references  are 
given,  upon  the  researches  of  Mittag-Leffler  and  Weierstrass.  The  most  important  investigations 
of  Mittag-Leffler  are  contained  in  a  series  of  short  notes,  constituting  the  memoir  "  Sur  la  th6orie 
des  fonctions  uniformes  d'une  variable,"  Comptes  Rendus,  t.  xciv,  (1882),  pp.  414,  511,  713,  781, 
938,  1040,  1105,  1163,  t.  xcv,  (1882),  p.  335  ;  and  in  a  memoir  "  Sur  la  representation  analytique 
des  fonctions  monogenes  uniformes,"  Acta  Math.,  t.  iv,  (1884),  pp.  1 — 79.  The  investigations  of 
Weierstrass  referred  to  are  contained  in  his  two  memoirs  "  Ueber  einen  functionentheoretischen 
Satz  des  Herrn  G.  Mittag-Leffler,"  (1880),  and  "  Zur  Functionenlehre,"  (1880),  both  included  in 
the  volume  Abhandlungen  aus  der  Functionenlehre,  pp.  53 — 66,  67 — 101,  102 — 104.  A  memoir  by 
Hermite  "  Sur  quelques  points  de  la  theorie  des  fonctions,"  Acta  Soc.  Fenn.,  t.  xii,  pp.  67 — 94, 
Crelle,  t.  xci,  (1881),  pp.  54 — 78  may  be  consulted  with  great  advantage. 


72.] 


MITTAG-LEFFLER'S  THEOREM 


113 


sole  singularity;  the  singularity  is  essential  or  accidental  according  as 
GI  is  transcendental  or  algebraical.  These  functions  can  be  constructed 
by  theorems  already  proved.  Then  we  have  the  theorem,  due  to  Mittag- 
Le  frier: — It  is  always  possible  to  construct  a  uniform  analytical  function  F  (z), 
having  no  singularities  other  than  a1}  a«,  a,,  ...  and  such  that  for  each  deter 
minate  value  of  v,  the  difference  F  (z)-Gv  ( )  is  finite  for  z  =  av  and 

\z     av/ 

therefore,  in  the  vicinity  of  av,  is  expressible  in  the  form  P  (z  —  «„). 

73.     To  prove  Mittag-Leffler's  theorem,  we  first  form  subsidiary  functions 

Fv  (z),  derived  from  the  functions  G  as  follows.     The  function  Gv  (—- — } 

\z  —  aj 

converges  everywhere  in  the  plane  except  at  the  point  «„;  hence  within  a 
circle  z  <  av\  it  is  a  monogenic  analytic  function  of  z,  and  can  therefore  be 
expanded  in  a  series  of  positive  powers  of  z  which  converges  uniformly 
within  the  circle,  say 


z-a 


for  values  of  z  such  that  \z\  <  av .     If  a,,  be  zero,  there  is  evidently  no 
expansion. 

Let  e  be  a  positive  quantity  less  than  1,  and  let  elf  e2,  e3,  ...  be  arbitrarily 
chosen  positive  decreasing  quantities,  subject  to  the  single  condition  that  2e 
is  a  converging  series,  say  of  sum  A  :  and  let  e0  be  a  positive  quantity  inter 
mediate  between  1  and  e.  Let  g  be  the  greatest  value  of  ~  f 


z  —  a, 


for 


points  on  or  within  the  circumference  \z\  =  e0  a,|;  then,  because  the  series 

00 

2  v^z*  is  a  converging  series,  we  have,  by  §  29, 


or 


Hence,  with  values  of  z  satisfying  the  condition  \z\^.e  av\,  we  have,  for 
any  value  of  m, 


/j.=m 


Vu      Z 


2,  q  - 

9  mJt 

n  =  m      fco 


1- 


since  e<e0.     Take  the  smallest  integral  value  of  m  such  that 

9 


F. 


114  MITTAG-LEFFLER'S 

it  will  be  finite  and  may  be  denoted  by  mv :  and  thus  we  have 


[73. 


for  values  of  z  satisfying  the  condition  \z\^.e  av\. 

We  now  construct  a  subsidiary  function  Fv  (z)  such  that,  for  all  values  of  z, 


then  for  values  of  UL  which  are  ^  e  aJ, 


Moreover,  the  function     2    zv^  is  finite  for  all  finite  values  of  z  so  that,  if  we 
n=o 

take 

.j 


-a 


—  i 


then  6,,(^)  is  zero  at  infinity,  because,  when  5=00,  #„(-     -)is  finite  by 

\z  —  civ/ 

hypothesis.     Evidently  <f>v(z)  is  infinite  only  at  z  =  av,  and  its  singularity  is 
of  the  same  kind  as  that  of  Gt 


z  —  a, 


74.  Now  let  c  be  any  point  in  the  plane,  which  is  not  one  of  the  points 
«],  a2,  as,  ...;  it  is  possible  to  choose  a  positive  quantity  p  such  that  no  one 
of  the  points  a  is  included  within  the  circle 


z  —  c 


=  p- 


Let  av  be  the  singularity,  which  is  the  point  nearest  to  the  origin  satisfy 
ing  the  condition    «„  >  c  \  +  p  ;  then,  for  points  within  or  on  the  circle,  we 

have 

'  z 

as 

when  s  has  the  values  v,  v  +  1,  v  +  2, Introducing  the  subsidiary  functions 

Fv  (z),  we  have,  for  such  values  of  z, 


and  therefore 


F.(z) 


a  finite  quantity.     It  therefore  follows  that  the  series   2  F,  (z)  converges  uni- 

8=v 

formly  and  unconditionally  for  all  values  of  z  which  satisfy  the  condition 


74.]  THEOREM  115 

z  —  c\^.p.  Moreover,  all  the  functions  Fl(z),  F2(z),  ...,  Fr_l(z]  are  finite  for 
such  values  of  z,  because  their  singularities  lie  without  the  circle  z  —  c  =  p  ; 
and  therefore  the  series 

S  Fr(z) 

r=l 

converges  uniformly  and  unconditionally  for  all  points  z  within  or  on  the 
circle  \z  —  c  =p,  where  p  is  chosen  so  that  the  circle  encloses  none  of  the 
points  a. 

The  function,  represented  by  the  series,  can  therefore  be  expanded  in  the 
form  P  (z  —  c),  in  the  domain  of  the  point  c. 

If  am  denote  any  one  of  the  points  a1}  a2,  ...,  and  we  take  p'  so  small  that 
all  the  points,  other  than  am,  lie  without  the  circle 

I  / 

I  *       U"m   —  P  ) 

then,  since  Fm  (z)  is  the  only  one  of  the  functions  F  which  has  a  singularity 
at  am,  the  series 

^{Fr(z}}-Fm(z) 

converges  regularly  in  the  vicinity  of  a,  and  therefore  it  can  be  expressed  in 
the  form  P  (z  —  am).  Hence 


a 


the  difference  of  Fm  and  Gm  being  absorbed  into  the  series  P  to  make  Pj.     It 

GO 

thus  appears  that  the  series  2  Fr  (z)  is  a  function  which  has  infinities  only 

r  =  \ 

at  the  points  a1}  a2,  ...,  and  is  such  that 


can  be  expressed  in  the  vicinity  of  am  in  the  form  P  (z  -  am).    Hence  2  Fr  (z) 
is  a  function  of  the  required  kind. 

75.  It  may  be  remarked  that  the  function  is  by  no  means  unique.  As 
the  positive  quantities  e  were  subjected  to  merely  the  single  condition  that 
they  form  a  converging  series,  there  is  the  possibility  of  wide  variation  in 
their  choice:  and  a  difference  of  choice  might  easily  lead  to  a  difference 
in  the  ultimate  expression  of  the  function. 

This  latitude  of  ultimate  expression  is  not,  however,  entirely  unlimited. 
For,  suppose  there  are  two  functions  F(z)  and  F  (z\  enjoying  all  the  assigned 
properties.  Then  as  any  point  c,  other  than  a^,  a2,  ...,  is  an  ordinary  point  for 
both  F  (z)  and  F  (z),  it  is  an  ordinary  point  for  their  difference  :  and  so 

F(z)-F(z)  =  P(z-c) 

8—2 


116  FUNCTIONS   POSSESSING  [75. 

for  points  in  the  immediate  vicinity  of  c.  The  points  a  are,  however, 
singularities  for  each  of  the  functions :  in  the  vicinity  of  such  a  point  a* 
we  have 


since  the  functions  are  of  the  required  form  :  hence 

F(z}-F(z}=P(z-ai)  -P(z-  ai), 

or  the  point  a;  is  an  ordinary  point  for  the  difference  of  the  functions.  Hence 
every  finite  point  in  the  plane,  whether  an  ordinary  point  or  a  singularity 
for  each  of  the  functions,  is  an  ordinary  point  for  the  difference  of  the 
functions  :  and  therefore  that  difference  is  a  uniform  integral  function  of  z. 
It  thus  appears  that,  if  F  (z)  be  a  function  with  the  required  properties,  then 
every  other  function  with  those  properties  is  of  the  form 

F(z)  +  G(z], 

where  G  (z)  is  a  uniform  integral  function  of  z  either  transcendental  or 
algebraical. 

The  converse  of  this  theorem  is  also  true. 

00 

Moreover,  the  function  G  (z)  can  always  be  expressed  in  a  form  2  gv(z),  if 

v=\ 

it  be  desirable  to  do  so  :  and  therefore  it  follows  that  any  function  with  the 
assigned  characteristics  can  be  expressed  in  the  form 


76.  The  following  applications,  due  to  Weierstrass,  can  be  made  so  as  to 
give  a  new  expression  for  functions,  already  considered  in  Chapter  VI.,  having 
z  =  oo  as  their  sole  essential  singularity  and  an  unlimited  number  of  poles  at 
points  Oi,  a2,  — 

If  the  pole  at  af  be  of  multiplicity  mi}  then  (z  —  a$n>f(z)  is  regular  at 
the  point  a;  and  can  therefore  be  expressed  in  the  form 


mi—  1 

Hence,  if  we  take  /f  (z)  =   2   c^  (z  —  ai)~TO<+'t, 

M  =  0 

we  have  f(z}  =fi  (z)  +  P  (z  —  «;). 

Now  deduce  from  fi(z)  a  function  Fi(z)  as  in  |  73,  and  let  this  deduction  be 

effected  for  each  of  the  functions  /,-  (z).     Then  we  know  that 


is  a  uniform  function  of  z  having  the  points  a1}  a2,  ...  for  poles  in  the  proper 


76.]  UNLIMITED   SINGULARITIES  117 

multiplicity  and  no  essential  singularity  except  z  =  oo  .     The  most  general 
form  of  the  function  therefore  is 


r=\ 

Hence  any  uniform  analytical  function  which  has  no  essential  singularity 
except  at  infinity  can  be  expressed  as  a  sum  of  functions  each  of  which  has  only 
one  singularity  in  the  finite  part  of  the  plane.  The  form  of  Fr  (z)  is 

fr(z}-Gr(z\ 

where  fr  (z)  is  infinite  at  z  =  ar  and  Gr  (z)  is   a   properly  chosen   integral 
function. 

We  pass  to  the  case  of  a  function  having  a  single  essential  singularity  at 
c  and  at  no  other  point  and  any  number  of  accidental  singularities,  by  taking 

z'  =  -  as  in  §  63.  Cor.  II.:  and  so  we  obtain  the  theorem  : 

z  —  c 

Any  uniform  function  which  has  only  one  essential  singularity,  which  is 
at  c,  can  be  expressed  as  a  sum  of  uniform  functions  each  of  which  has  only 
one  singularity  different  from  c. 

Evidently  the  typical  summative  function  Fr  (z)  for  the  present  case  is  of 
the  form 


Z  — 


77.  The  results,  which  have  been  obtained  for  functions  possessed  of 
an  infinitude  of  singularities,  are  valid  on  the  supposition,  stated  in  §  72, 
that  the  limit  of  av  with  indefinite  increase  of  v  is  infinite ;  the  series 
ttj,  «2,  •••  tends  to  one  definite  limiting  point  which  is  2=00  and,  by  the 
substitution  z'  (z  —  c)  =  1,  can  be  made  any  point  c  in  the  finite  part  of  the 
plane. 

Such  a  series,  however,  does  not  necessarily  tend  to  one  definite  limiting 
point:  it  may,  for  instance,  tend  to  condensation  on  a  curve,  though  the 
condensation  does  not  imply  that  all  points  of  the  continuous  arc  of  the  curve 
must  be  included  in  the  series.  We  shall  not  enter  into  the  discussion 
of  the  most  general  case,  but  shall  consider  that  case  in  which  the  series  of 
moduli  \al)  a2  ,  ...  tends  to  one  definite  limiting  value  so  that,  with  in 
definite  increase  of  v,  the  limit  of  \av  is  finite  and  equal  to  R ;  the  points 
«i,  «2,  ...  tend  to  condense  on  the  circle  \z  =  R. 
Such  a  series  is  given  by 

2fori 
(        I  _  l  -m+n 

«„,*={!  + 


for  &=0,  1,  ...,  n,  and  n=l,  2,  ...  ad  inf.;  and  another*  by 

a«Hl  +  (-l)ncn}e2M7"V2, 
where  c  is  a  positive  proper  fraction. 

*  The  first  of  these  examples  is  given  by  Mittag-Leffler,  Acta  Math.,  t.  iv,  p.  11 ;  the  second 
was  stated  to  me  by  Mr  Burnside. 


118  FUNCTIONS   POSSESSING  [77. 

With  each  point  am  we  associate  the  point  on  the  circumference  of  the 
circle,  say  bm,  to  which  am  is  nearest:  let 

|  dm       "m  I  =  Pm> 

so  that  pm  approaches  the  limit  zero  with  indefinite  increase  of  m.  There 
cannot  be  an  infinitude  of  points  ap,  such  that  pp^<&,  any  assigned  positive 
quantity ;  for  then  either  there  would  be  an  infinitude  of  points  a  within  or 
on  the  circle  \z\  =  R  —  ®,  or  there  would  be  an  infinitude  of  points  a  within 
or  on  the  circle  z  =  R  +  ©,  both  of  which  are  contrary  to  the  hypothesis 
that,  with  indefinite  increase  of  v,  the  limit  of  \av  is  R.  Hence  it  follows 
that  a  finite  integer  n  exists  for  every  assigned  positive  quantity  ®,  such  that 

\am-bm\  <  ® 
when  m^n. 

Then  the  theorem,  which  corresponds  to  Mittag-LefHer's  as  stated  in  §  72 
and  which  also  is  due  to  him,  is  as  follows : — 

It  is  always  possible  to  construct  a  uniform  analytical  function  of  z  which 
exists  over  the  whole  plane,  except  at  the  points  a  and  b,  and  which,  in  the 
immediate  vicinity  of  each  one  of  the  singularities  a,  can  be  expressed  in  the  form 


where  the  functions  G{  are  assigned  functions,  vanishing  with  -     -  and  finite 

Z  —  (Li 

everywhere  in  the  plane  except  at  the  single  points  a;  with  which  they  are 
respectively  associated. 

In  establishing  this  theorem,  we  shall  need  a  positive  quantity  e  less  than 
unity  and  a  converging  series  e^  e2,  e3,  ...  of  positive  quantities,  all  less  than 
unity. 

Let  the  expression  of  the  function  Gn  be 

"I       /  _.  ..       \0       I       /  -  _.       \5       '        '  '  '    ' 


n  \z  -  a     ~  z-an     (z-  an)2     (z  -  an)s 

Then,  since  z  -  an  =  (z  -  bn)  \l  --  —  ~l\  , 

(        z     on  ) 

the  function  Gn  can  be  expressed*  in  the  form 


l«— <li 

for  values  of  z  such  that 


an  - 


z-bn 
and  the  coefficients  A  are  given  by  the  equations 


*  The  justification  of  this  statement  is  to  be  found  in  the  proposition  in  §  82. 


77.] 


UNLIMITED   SINGULARITIES 


119 


Now,  because  Gn  is  finite  everywhere  in  the  plane  except  at  an,  the  series 


has  a  finite  value,  say  #,  for  any  non-zero  value  of  the  positive  quantity  %n ; 
then 


Hence 


0*-!)! 


ft  &  f 

<    S   flfr-^ 


71  ?^ 

Introducing  a  positive  quantity  a  such  that 


we  choose  £n  so  that  £n  <  a|an  -  bn\ ; 

and  then  |  A  n>  ^  \  <  go.  ( 1  +  a)*-1. 

Because  (1  +  a)  e  is  less  than  unity,  a  quantity  6  exists  such  that 

(1  +  a)  e  <  6  <  1. 


Then  for  values  of  z  determined  by  the  condition 

go.    6 


dn     on 


<  e,  we  have 


al-0' 


Let  the  integer  mn  be  chosen  so  that 

ga     &> 


it  will  be  a  finite  integer,  because  0<  1.     Then 

00  (1       7) 

V  I   A  I      "H         ^^ 


We  now  construct,  as  in  §  73,  a  subsidiary  function  Fn(z),  defining  it  by 
the  equation 


so  that  for  points  z  determined  by  the  condition 

\Fn(z)\<en. 
A  function  with  the  required  properties  is 

00 

Fm(z\ 


<  €,  we  have 


m=l 


120  FUNCTIONS   POSSESSING  [77. 

To  prove  it,  let  c  be  any  point  in  the  plane  distinct  from  any  of  the  points 
a  and  b ;  we  can  always  find  a  value  of  p  such  that  the  circle 

\z-c\=p 

contains  none  of  the  points  a  and  b.     Let  I  be  the  shortest  distance  between 
this  circle  and  the  circle  of  radius  R,  on  which  all  the  points  b  lie ;  then  for 


all  points  z  within  or  on  the  circle 


z  —  c 


—  p  we  have 


Now  we  have  seen  that,  for  any  assigned  positive  quantity  <s),  there  is  a 
finite  integer  n  such  that 

I  dm  —  bm  <  © 

when  m  ^  n.     Taking  ®  =  el,  we  have 

m 

<  e 


when  m^n,n  being  the  finite  integer  associated  with  the  positive  quantity  el. 
It  therefore  follows  that,  for  points  z  within  or  on  the  circle  \z  —  c\  =  p, 

\Fm(z}\<em, 
when  m  is  not  less  than  the  finite  integer  n.     Hence 


a  finite  quantity  because  e1}  e2,  ...  is  a  converging  series;  and  therefore 


is  a  converging  series.     Each  of  the  functions  F1(z),  F»(z),  ...,  Fn_-i(z)  is 
finite  when    z  —  c  ^  p  ;   and  therefore 


is  a  series  which  converges  uniformly  and  unconditionally  for  all  values  of  z 
included  in  the  region 

\z-c\^p. 

Hence  the  function  represented  by  the  series  can  be  expressed  in  the  form 
P  (z  —  c)  for  all  such  values  of  z.  The  function  therefore  exists  over  the 
whole  plane  except  at  the  points  a  and  b. 

It  may  be  proved,  exactly  as  in  §  74,  that,  for  points  z  in  the  immediate 
vicinity  of  a  singularity  am, 


The  theorem  is  thus  completely  established. 

The  function  thus  obtained  is  not  unique,  for  a  wide  variation  of  choice  of 
the  converging  series  ea  +  e2  +  . . .  is  possible.     But,  in  the  same  way  as  in  the 


77.] 


UNLIMITED   SINGULARITIES 


121 


corresponding  case  in  §  75,  it  is  proved  that,  if  F  (z)  be  a  function  with  the 
required  properties,  every  other  function  with  those  properties  is  of  the  form 

F(z}+G(z\ 

where  G  (z)  behaves  regularly  in  the  immediate  vicinity  of  every  point  in  the 
plane  except  the  points  b. 

78.  The  theorem  just  given  regards  the  function  in  the  light  of  an 
infinite  converging  series  of  functions  of  the  variable  :  it  is  natural  to  suppose 
that  a  corresponding  theorem  holds  when  the  function  is  expressed  as  an 
infinite  converging  product.  With  the  same  series  of  singularities  as  in 
§  77,  when  the  limit  of  av  with  indefinite  increase  of  v  is  finite  and 
equal  to  R,  the  theorem*  is:  — 

It  is  always  possible  to  construct  a  uniform  analytical  function  which 
behaves  regularly  everywhere  in  the  plane  except  at  the  points  a  and  b  and 
which  in  the  vicinity  of  any  one  of  the  points  av  can  be  expressed  in  the 
form 


where  the  numbers  w1}  n2,  ...  are  any  assigned  integers. 

The  proof  is  similar  in  details  to  proofs  of  other  propositions  and  it  will 
therefore  be  given  only  in  outline.     We  have 


au- 


provided 


such  values  of  z, 


z-av     z-bv     z  -  bv  ^i  V  z  -  bv  J  ' 
<  e,  the  notation  being  the  same  as  in  §  77.     Hence,  for 


=e 


(/7    _  7)  \ 
i_  ^  _  M 
2-bJ 


-n,,     S 


by  Ev  (z),  we  have  Ev  (z}  =e          m" 

Hence,  if  F(z)  denote  the  infinite  product 


we  have  F(z)  =  e 

and  F(z)  is  a  determinate  function  provided  the  double  series  in  the  index  of 

the  exponential  converge. 

*  Mittag-Leffler,  Acta  Math.,  t.  iv,  p.  32  ;  it  may  be  compared  with  Weierstrass's  theorem  in 
§67. 


122  TRANSCENDENTAL   FUNCTION   AS 

Because  nv  is  a  finite  integer  and  because 


[78. 


is  a  converging  series,  it  is  possible  to  choose  an  integer  mv  so  that 

7) 

"x 


M(T^ 


where  t]v  is  any  assigned  positive  quantity.  We  take  a  converging  series  of 
positive  quantities  rjv :  and  then  the  moduli  of  the  terms  in  the  double  series 
form  a  converging  series.  The  double  series  itself  therefore  converges 
uniformly  and  unconditionally ;  and  then  the  infinite  product  F  (z)  converges 
uniformly  and  unconditionally  for  points  z  such  that 


&„  —  b.. 


<  e. 


As  in  §  77,  let  c  be  any  point  in  the  plane,  distinct  from  any  of  the 
points  a  and  b.  We  take  a  finite  value  of  p  such  that  the  circle  z  —  c\=p 
contains  none  of  the  points  a  and  b ;  and  then,  for  all  points  within  or  on  this 
circle, 


z— 


<e 


when  m^n,  n  being  the  finite  integer  associated  with  the  positive  quantity 
el.     The  product 

fi  Ev(z) 

v=n 

is  therefore  finite,  for  its  modulus  is  less  than 

CO 

S    IJK 

K  =  » 


the  product 


n 

v=l 


is  finite,  because  the  circle  z  —  c\  =  p  contains  none  of  the  points  a  and  6; 
and  therefore  the  function  F(z)  is  finite  for  all  points  within  or  on  the  circle. 
Hence  in  the  vicinity  of  c,  the  function  can  be  expanded  in  the  form  P  (z  —  c)  ; 
and  therefore  the  function  exists  everywhere  in  the  plane  except  at  the  points 
a  and  b. 

The  infinite  product  converges ;  it  can  be  zero  only  at  points  which  make 
one  of  the  factors  zero  and,  from  the  form  of  the  factors,  this  can  take  place 
only  at  the  points  av  with  positive  integers  nv.  In  the  vicinity  of  av  all 
the  factors  of  F  (z)  except  Ev  (z)  are  regular ;  hence  F  (z)\Ev  (z)  can  be 
expressed  as  a  function  of  z  —  av  in  the  vicinity.  But  the  function  has  no 
zeros  there,  and  therefore  the  form  of  the  function  is 

Pl  (z-a,,). 


78.]  AN   INFINITE   SERIES   OF   FUNCTIONS  123 

Hence  in  the  vicinity  of  av,  we  have 


on  combining  with  Pl  (z  —  av)  the  exponential  index  in  Ev(z).     This  is  the 
required  property. 

Other  general  theorems  will  be  found  in  Mittag-Leffler's  memoir  just 
quoted. 

79.  The  investigations  in  §§  72  —  75  have  led  to  the  construction  of  a 
function  with  assigned  properties.  It  is  important  to  be  able  to  change,  into 
the  chosen  form,  the  expression  of  a  given  function,  having  an  infinite  series 
of  singularities  tending  to  a  definite  limiting  point,  say  to  z  =  oo  .  It  is 
necessary  for  this  purpose  to  determine  (i)  the  functions  Fr(z)  so  that  the 

00 

series    2  Fr  (z)  may  converge  uniformly  and  (ii)  the  function  G  (z). 

r=l 

Let  <&  (z)  be  the  given  function,  and  let  S  be  a  simple  contour  embracing 
the  origin  and  /j,  of  the  singularities,  viz.,  al  ,  ......  ,  aM:   then,  if  t  be  any 

point,  we  have 

-          «  «  . 


m  r«  *£)  ,,y         r«  *y)  ,,. 

J      t-z\t)  J     t-z\t) 


f(a)  _ 

where  I      implies  an  integral  taken  round  a  very  small  circle  centre  a. 

If  the  origin  be  one  of  the  points  a1}  a2,  ......  ,  then  the  first  term  will  be 

included  in  the  summation. 

Assuming  that  z  is  neither  the  origin  nor  any  one  of  the  points  a1}  ...,  a^, 
we  have 


so 


27TI 


AT  ^ 

Now     —  .       7-^-7      dt 


1    [(0)$>(t)fz\ 
—  .       7-^-7 
Ziri]      t-z\tj 


, 
—  -—.  2  I      7—^-    -      dt. 

t-Z\t) 


(ffl-l)i  I 

\~dm  1(®(t)  +  ^i^+^ 


[ 


124  TRANSCENDENTAL   FUNCTION   AS  [79. 

unless  z  =  0  be  a  singularity  and  then  there  will  be  no  term  G  (z).     Similarly, 
it  can  be  shewn  that 


/       I        \         m-l         /  z  \  A 

is  equal  to  Gv(-      -}  -  2  vj-}  =  F,  (z), 

\z  -  aj      A=0      \aj 


where  ,  —     s—  • 

2?rt 

and  the  subtractive  sum  of  m  terms  is  the  sum  of  the  first  m  terms  in  the 
development  of  Gv  in  ascending  powers  of  z.     Hence 


If,  for  an  infinitely  large  contour,  m  can  be  chosen  so  that  the  integral 


t- 


diminishes  indefinitely  with  increasing  contours  enclosing  successive  singu 
larities,  then 


The  integer  m  may  be  called  the  critical  integer. 
If  the  origin  be  a  singularity,  we  take 


and  there  is  then  no  term  G  (z)  :  hence,  including  the  origin  in  the  summa 
tion,  we  then  have 


so  that  if,  for  this  case  also,  there  be  some  finite  value  of  m  which  makes 
the  integral  vanish,  then 


Other  expressions  can  be  obtained  by  choosing  for  m  a  value  greater  than 
the  critical  integer  ;  but  it  is  usually  most  advantageous  to  take  m  equal  to 
its  least  lawful  value. 

Ex.  1.     The  singularities  of  the  function  ?r  cot  772  are  given  by  z  =  \,  for  all  integer 
values  of  X  from  —  oo  to  +00  including  zero,  so  that  the  origin  is  a  singularity. 

The  integral  to  be  considered  is 


-      1     M  IT  cot  vt  fz\m  ,, 
=  ~ — .  I     — -  (-  )    at. 

2iri  J          t-z      \tj 


We  take  the  contour  to  be  a  circle  of  very  large  radius  R  chosen  so  that  the  circumference 
does  not  pass  innuitesimally  near  any  one  of  the  singularities  of  TT  coint  at  infinity;  this 


79.]  AN    INFINITE   SERIES   OF   FUNCTIONS  125 

is,  of  course,  possible  because  there  is  a  finite  distance  between  any  two  of  them.  Then, 
round  the  circumference  so  taken,  n  cot  nt  is  never  infinite  :  hence  its  modulus  is  never 
greater  than  some  finite  quantity  M. 

Let  t  =  Reei,  so  that  ~=id6;  then 

v 


and  therefore 


Z 


.--—. 
t-z 


for  some  point  t  on  the  circle.     Now,  as  the  circle  is  very  large,  we  have  \t-z\  infinite  : 
hence  \J\  can  be  made  zero  merely  by  taking  m  unity. 

Thus,  for  the  function  TT  cot  TTZ,  the  critical  integer  is  unity. 
Hence  from  the  general  theorem  we  have  the  equation 


1         fir  cot  nt  z  j 

7T  COt  772=  -5— .  2     I— -dt, 

2TTI         J        t-Z        t 


the  summation  extending  to  all  the  points  X  for  integer  values  of  X  =  -  oc  to  +  oo  ,  and 
each  integral  being  taken  round  a  small  circle  centre  X. 

-vr         .  »    .  1      /"(*)  TT  cot  irt  z   , 

Now  if,  in  -—  .  •  -dt. 

2m  J         t  -  z     t 

we  take  t=\  +  (,  we  have 


where  P(Q  =  0  when  £=  0;  and  therefore  the  value  of  the  integral  is 


•*./ (*-*+{)  (x+fl  t 

In  the  limit  when  |f|  is  infinitesimal,  this  integral 

z 

=  (X-2)X 

1      1 

~X-2        X' 

and  therefore  /*.  (z)  =  -J—  +  1 

A    '     z-X      X' 

if  X  be  not  zero. 

And  for  the  zero  of  X,  the  value  of  the  integral  is 


(p 


126  REGION   OF   CONTINUITY  [79. 

so  that  F0(z)  is  -.     In  fact,  in  the  notation  of  §  72,  we  have 

z 


o  P-A»JL 

^  \z-\J~z-\' 
arid  the  expansion  of  GK  needs  to  be  carried  only  to  one  term. 

1       A=ao  /I        1\ 
We  thus  have  7rcot7rs  =  — f-     2        — N+=r)> 

z       A=-co  \Z-X       A/ 

the  summation  not  including  the  zero  value  of  X. 
Ex.  2.     Obtain,  ab  initio,  the  relation 


SHI2  3       A=_aj  (z-X7r)2' 

p.  3.     Shew  that,  if 


1  °°      1         1 

then  "-^^  =  -  +  2z  2  ^3-^1- 

R(z)        z         i=lR(\)z*-\* 

(Gylden,  Mittag-Leffler.) 

Ex.  4.     Obtain  an  expression,  in  the  form  of  a  sum,  for 

IT  cot  irz 


where  Q(z)  denotes  (1  -z)  (l  -^  (l  -|J  ......  ^-j)*- 


80.  The  results  obtained  in  the  present  chapter  relating  to  functions 
which  have  an  unlimited  number  of  singularities,  whether  distributed  over 
the  whole  plane  or  distributed  over  only  a  finite  portion  of  it,  shew  that 
analytical  functions  can  be  represented,  not  merely  as  infinite  converging 
series  of  powers  of  the  variable,  but  also  as  infinite  converging  series  of 
functions  of  the  variable.  The  properties  of  functions  when  represented  by 
series  of  powers  of  the  variable  depended  in  their  proof  on  the  condition  that 
the  series  proceeded  in  powers;  and  it  is  therefore  necessary  at  least  to 
revise  those  properties  in  the  case  of  functions  when  represented  as  series 
of  functions  of  the  variable. 

Let  there  be  a  series  of  uniform  functions  /i  (z),  /,  (z),  .  .  .  ;  then  the 
aggregate  of  values  of  z,  for  which  the  series 


1*1 

has  a  finite  value,  is  the  region  of  continuity  of  the  series.     If  a  positive 
quantity  p  can  be  determined  such  that,  for  all  points  z  within  the  circle 

z  —  a\  =  p, 


80.]  OF   A    SERIES    OF   FUNCTIONS  127 

00 

the  series  2  fi(z)  converges  uniformly  and  unconditionally*,  the  series  is 

said  to  converge  in  the  vicinity  of  a.     If  R  be  the  greatest  value  of  p  for 
which  this  holds,  then  the  area  within  the  circle 

z  —  a\  =  R 

is  called  the  domain  of  a;  and  the  series  converges  uniformly  and  uncon 
ditionally  in  the  vicinity  of  any  point  in  the  domain  of  a. 

It  will  be  proved  in  §  82  that  the  function  can  be  represented  by  power- 
series,  each  such  series  being  equivalent  to  the  function  within  the  domain  of 
some  one  point.  In  order  to  be  able  to  obtain  all  the  power-series,  it  is 
necessary  to  distribute  the  region  of  continuity  of  the  function  into  domains 
of  points  where  it  has  a  uniform,  finite  value.  We  therefore  form  the  domain 
of  a  point  6  in  the  domain  of  a  from  a  knowledge  of  the  singularities  of  the 
function,  then  the  domain  of  a  point  c  in  the  domain  of  6,  and  so  on ;  the 
aggregate  of  these  domains  is  a  continuous  part  of  the  plane  which  has 
isolated  points  and  which  has  one  or  several  lines  for  its  boundaries.  Let 
this  part  be  denoted  by  At. 

For  most  of  the  functions,  which  have  already  been  considered,  the  region 
A1}  thus  obtained,  is  the  complete  region  of  continuity.  But  examples  will 
be  adduced  almost  immediately  to  shew  that  A-^  does  not  necessarily  include 
all  the  region  of  continuity  of  the  series  under  consideration.  Let  a'  be  a 
point  not  in  A-^  within  whose  vicinity  the  function  has  a  uniform,  finite 
value ;  then  a  second  portion  A2  can  be  separated  from  the  whole  plane,  by 
proceeding  from  a'  as  before  from  a.  The  limits  of  A±  and  A2  may  be  wholly 
or  partially  the  same,  or  may  be  independent  of  one  another :  but  no  point 
within  either  can  belong  to  the  other.  If  there  be  points  in  the  region  of  con 
tinuity  which  belong  to  neither  A1  nor  A2,  then  there  must  be  at  least  another 
part  of  the  plane  A3  with  properties  similar  to  At  and^l2-  And  so  on.  The 

00 

series  2  fi(z)  converges  uniformly  and  unconditionally  in  the  vicinity  of 

»=i  « 

every  point  in  each  of  the  separate  portions  of  its  region  of  continuity. 

It  was  proved  that  a  function  represented  by  a  series  of  powers  has  a 
definite  finite  derivative  at  every  point  lying  actually  within  the  circle 
of  convergence  of  the  series,  but  that  this  result  cannot  be  affirmed  for  a 
point  on  the  boundary  of  the  circle  of  convergence  even  though  the  value  of 
the  series  itself  should  be  finite  at  the  point,  an  illustration  being  provided 
by  the  hypergeometric  series  at  a  point  on  the  circumference  of  its  circle  of 

*  In  connection  with  most  of  the  investigations  in  the  remainder  of  this  chapter,  Weierstrass's 
memoir  "  Zur  Functionenlehre  "  already  quoted  (p.  112,  note)  should  be  consulted. 

It  may  be  convenient  to  give  here  Weierstrass's  definition  (I.e.,  p.  70)  of  uniform,  unconditional 


convergence.    A  series  2  fn  converges  uniformly,  if  an  integer  m  can  be  determined  so  that 


/» 


can  be  made  less  than  any  arbitrary  positive  quantity,  however  small ;  and  it  converges  uncon 
ditionally,  if  the  uniform  convergence  of  the  series  be  independent  of  any  special  arrangement 
of  order  or  combination  of  the  terms. 


128  REGION   OF   CONTINUITY   OF  [80. 

convergence.  It  will  appear  that  a  function  represented  by  a  series  of 
functions  has  a  definite  finite  derivative  at  every  point  lying  actually  within 
its  region  of  continuity,  but  that  the  result  cannot  be  affirmed  for  a  point 
on  the  boundary;  and  an  example  will  be  given  (§  83)  in  which  the  derivative 
is  indefinite. 

Again,  it  has  been  seen  that  a  function,  initially  defined  by  a  given  power- 
series,  is,  in  most  cases,  represented  by  different  analytical  expressions  in 
different  parts  of  the  plane,  each  of  the  elements  being  a  valid  expression  of 
the  function  within  a  certain  region.  The  questions  arise  whether  a  given 
analytical  expression,  either  a  series  of  powers  or  a  series  of  functions : 
(i)  can  represent  different  functions  in  the  same  continuous  part  of  its  region 
of  continuity,  (ii)  can  represent  different  functions  in  distinct  (that  is,  non- 
continuous)  parts  of  its  region  of  continuity. 

81.     Consider  first  a  function  defined  by  a  given  series  of  powers. 

Let  there  be  a  region  A'  in  the  plane  and  let  the  region  of  continuity  of 
the  function,  say  g  (z),  have  parts  common  with  A'.  Then  if  a0  be  any  point 
in  one  of  these  common  parts,  we  can  express  g  {z)  in  the  form  P  (z  —  a0)  in 
the  domain  of  a0. 

As  already  explained,  the  function  can  be  continued  from  the  domain  of 
a0  by  a  series  of  elements,  so  that  the  whole  region  of  continuity  is  gradually 
covered  by  domains  of  successive  points ;  to  find  the  value  in  the  domain  of 
any  point  a,  it  is  sufficient  to  know  any  one  element,  say,  the  element  in  the 
domain  of  a0.  The  function  is  the  same  through  its  region  of  continuity. 

Two  distinct  cases  may  occur  in  the  continuations. 

First,  it  may  happen  that  the  region  of  continuity  of  the  function  g  (z) 
extends  beyond  A'.  Then  we  can  obtain  elements  for  points  outside  A', 
their  aggregate  being  a  uniform  analytical  function.  The  aggregate  of 
elements  then  represents  within  A'  a  single  analytical  function :  but  as  that 
function  has  elements  for  points  without  A,  the  aggregate  within  A'  does 
not  completely  represent  the  function.  Hence 

If  a  function  be  defined  within  a  continuous  region  of  a  plane  by  an 
aggregate  of  elements  in  the  form  of  power-series,  which  are  continuations  of 
one  another,  the  aggregate  represents  in  that  part  of  the  plane  one  (and  only 
one)  analytical  function :  but  if  the  power-series  can  be  continued  beyond  the 
boundary  of  the  region,  the  aggregate  of  elements  within  the  region  is  not  the 
complete  representation  of  the  analytical  function. 

This  is  the  more  common  case,  so  that  examples  need  not  be  given. 

Secondly,  it  may  happen  that  the  region  of  continuity  of  the  function  does 
not  extend  beyond  A'  in  any  direction.  There  are  then  no  elements  of  the 
function  for  points  outside  A'  and  the  function  cannot  be  continued  beyond 
the  boundary  of  A.  The  aggregate  of  elements  is  then  the  complete 
representation  of  the  function  and  therefore : 


81.]  A   SERIES   OF   POWERS  129 

If  a  function  be  defined  within  a  continuous  region  of  a  plane  by  an 
aggregate  of  elements  in  the  form  of  power-series,  which  are  continuations  of 
one  another,  and  if  the  power-series  cannot  be  continued  across  the  boundary  of 
that  region,  the  aggregate  of  elements  in  the  region  is  the  complete  representa 
tion  of  a  single  uniform  monogenic  function  which  exists  only  for  values  of  the 
variable  within  the  region. 

The  boundary  of  the  region  of  continuity  of  the  function  is,  in  the  latter 
case,  called  the  natural  limit  of  the  function*,  as  it  is  a  line  beyond  which 
the  function  cannot  be  continued.  Such  a  line  arises  for  the  series 

l  +  2z  +  ^  +  2z9  +  ...  , 

in  the  circle  \z  =  1,  a  remark  due  to  Kronecker;  other  illustrations  occur  in 
connection  with  the  modular  functions,  the  axis  of  real  variables  being  the 
natural  limit,  and  in  connection  with  the  automorphic  functions  (see  Chapter 
XXII.)  when  the  fundamental  circle  is  the  natural  limit.  A  few  examples 
will  be  given  at  the  end  of  the  present  Chapter. 

It  appears  that  Weierstrass  was  the  first  to  announce  the  existence  of  natural  limits 
for  analytic  functions,  Berlin  Monatsber.  (1866),  p.  617 ;  see  also  Schwarz,  Ges.  Werke, 
t.  ii,  pp.  240 — 242,  who  adduces  other  illustrations  and  gives  some  references ;  Klein  and 
Fricke,  Vorl.  uber  die  Theorie  der  elliptischen  Modulfunctioncn,  t.  i,  (1890),  p.  110;  Jordan, 
Cows  d' Analyse,  t.  iii,  pp.  609,  610.  Some  interesting  examples  and  discussions  of 
functions,  which  have  the  axis  of  real  variables  for  a  natural  limit,  are  given  by  Hankel, 
"  Untersuchungen  liber  die  unendlich  oft  oscillirenden  und  unstetigen  Functionen," 
Math.  Ann.,  t.  xx,  (1870),  pp.  63—112. 

82.     Consider  next  a  series  of  functions  of  the  variable ;  let  it  be 


The  region  of  continuity  may  be  supposed  to  consist  of  several  distinct  parts, 
in  the  most  general  case ;  let  one  of  them  be  denoted  by  A.  Take  some 
point  in  A,  say  the  origin,  which  is  either  an  ordinary  point  or  an  isolated 
singularity;  and  let  two  concentric  circles  of  radii  R  and  R'  be  drawn  in  A, 
so  that 

R  <  z  =r<R, 

and  the  space  between  these  circles  lies  within  A.  In  this  space,  each  term 
of  the  series  is  finite  and  the  whole  series  converges  uniformly  and  uncon 
ditionally. 

Now  let  fi  (z)  be  expanded  in  a  series  of  powers  of  z,  which  series  con 
verges  within   the  space  assigned,  and  in  that  expansion  let  ^  be  the  co- 

oo 

efficient  of  z* ;  then  we  can  prove  that  2  i^  is  finite  and  that  the  series 

( /  °°      \ 

s  |(sO 

n.     (\i  =  0       I 

*  Die  natiirliche  Grenze,  according  to  German  mathematicians. 
F. 


130  REGION   OF   CONTINUITY  [82. 

converges  uniformly  and  unconditionally  within  this  space,  so  that 

•x.  (/    oo 

2  /,(*)  =  2      2 


i=l  "  /A    {\i=Q 

00 

Because    the    infinite   series   2  fi  (z)   converges   uniformly   and   uncon 
ditionally,  a  number  n  can  be  chosen  so  that 


where  &  is  an  arbitrary  finite  quantity,  ultimately  made  infinitesimal;  and 
therefore  also 


i=n 

where  n'  >  n  and  is  infinite  in  the  limit.     Now  since  the  number  of  terms  in 
the  series 


is  not  infinite  before  the  limit,  we  have 


But  the  original  series  converges  unconditionally,  and  therefore  k  is  not  less 

n 

than  the  greatest  value  of  the  modulus  of  2  fi(z)  for  points  within  the 

i=n 

region;  hence,  by  §  29,  we  have 

n 

2  V   <  AT  <i. 
»•=» 

00 

Moreover,  A;  is  not  less  than  the  greatest  value  of  the  modulus  of  2  fi(z) 
in  the  given  region ;  and  so 

00 

2  i^  <  AT  *. 

i=n 

Now,  by  definition,  k  can  be  made  as  small  as  we  desire  by  choice  of  n ;  hence 

the  series 


is  a  converging  series.     Let  it  be  denoted  by  A^. 

n-l  oo 

Let  2  r'M  =  A  /,  2  ifj,  =  A  M" ; 

then,  by  the  above  suppositions,  we  can  always  choose  n  so  that 


k  being  any  assignable  small  quantity. 


82.] 


OF   A   SERIES   OF   FUNCTIONS 


131 


When  two  new  quantities  r±  and  r2  are  introduced,  as  in  §  28,  satisfying 
the  inequalities 

f-f    ^   ly     ^    \  iv       --•   /y»      ^     7?' 

-il/<^/l<s.|.S|<i./2<.-fl, 

the  integer  w  can  be  chosen  so  that 

\Ap'\  <  kr~*  <  kr^. 

f-  r. 


Then 


and 


so  that 


2 

.— 

00 

2 


- 


-     <k 


M=-oo  r  —  r-i        r2-r 

Hence   the   series     2   A^'z^   can   by  choice  of  n  be   made  to   have   a 
modulus  less  than  any  finite  quantity ;  and  therefore,  since 

/u.=  oo  n  —  1 

(for  there  is  a  finite  number  of  terms  in  the  coefficients  on  each  side,  the 
expansions  are  converging  series,  and  the  sum  on  the  right-hand  side  is  a 
finite  quantity),  it  follows  that  the  series 


converges  uniformly. 
Finally,  we  have 


2    . 

fl=  —00 


2  ft (*)  -  24^  =  2  /<  (z)  - 
<=i  1=1 


and  therefore 


2 

t'=n 


r  ~ 


which,  as  k  can  be  diminished  indefinitely,  can  be  made  less  than  any  finite 

jlX=00 

quantity.     Hence  the  series     2   A^  converges  unconditionally,  and  there- 

fi=  —00 

fore  we  have 


provided 


00  jlt=00 

2  /;(*)=      2    . 

l'=l  /u=  —  oo 


9—2 


132  REGION   OF   CONTINUITY  [82. 

When  we  take  into  account  all  the  parts  of  the  region  of  continuity 
of  the  series,  constituted  by  the  sum  of  the  functions,  we  have  similar 
expansions  in  the  form  of  successive  series  of  powers  of  z  —  c,  converging 
uniformly  and  unconditionally  in  the  vicinities  of  the  successive  points  c. 
But,  in  forming  the  domains  of  these  points  c,  the  boundary  of  the  region  of 
continuity  of  the  function  must  not  be  crossed ;  and  a  new  series  of  powers  is 
required  when  the  circle  of  convergence  of  any  one  series  (lying  within  the 
region  of  continuity)  is  crossed. 

It  therefore  appears  that  a  converging  series  of  functions  of  a  variable 
can  be  expressed  in  the  form  of  series  of  powers  of  the  variable  which 
converge  within  the  parts  of  the  plane  where  the  series  of  functions 
converges  uniformly  and  unconditionally ;  but  the  equivalence  of  the  two 
expressions  is  limited  to  such  parts  of  the  plane  and  cannot  be  extended 
beyond  the  boundary  of  the  region  of  continuity  of  the  series  of  functions. 

If  the  region  of  continuity  of  a  series  of  functions  consist  of  several  parts 
of  the  plane,  then  the  series  of  functions  can  in  each  part  be  expressed  in 
the  form  of  a  set  of  converging  series  of  powers :  but  the  sets  of  series  of 
powers  are  not  necessarily  the  same  for  the  different  parts,  and  they  are  not 
necessarily  continuations  of  one  another,  regarded  as  power-series. 

Suppose,  then,  that  the  region  of  continuity  of  a  series  of  functions 

F(z)=lfi(z) 

i=l 

consists   of  several   parts   A1}  A.2, Within   the   part    A^   let   F  (z)   be 

represented,  as  above,  by  a  set  of  power-series.  At  every  point  within  A1} 
the  values  of  F(z)  and  of  its  derivatives  are  each  definite  and  unique ;  so 
that,  at  every  point  which  lies  in  the  regions  of  convergence  of  two  of  the 
power-series,  the  values  which  the  two  power-series,  as  the  equivalents  of  F  (z) 
in  their  respective  regions,  furnish  for  F  (z)  and  for  its  derivatives  must  be 
the  same.  Hence  the  various  power-series,  which  are  the  equivalents  of  F  (z) 
in  the  region  Aly  are  continuations  of  one  another:  and  they  are  sufficient  to 
determine  a  uniform  monogenic  analytic  function,  say  F^  (z}.  The  functions 
F(z)  and  Fl(z)  are  equivalent  in  the  region  Al;  and  therefore,  by  §  81,  the 
series  of  functions  represents  one  and  the  same  function  for  all  points  within 
one  continuous  part  of  its  region  of  continuity.  It  may  (and  frequently  does) 
happen  that  the  region  of  continuity  of  the  analytical  function  F±  (z)  extends 
beyond  A± ;  and  then  F-^  (z)  can  be  continued  beyond  the  boundary  of  A^  by 
a  succession  of  elements.  Or  it  may  happen  that  the  region  of  continuity 
of  Fl  (z)  is  completely  bounded  by  the  boundary  of  A^ ;  and  then  the  function 
cannot  be  continued  across  that  boundary.  In  either  case,  the  equivalence 

00 

of  F-L(Z)   and    2  fi(z)  does  not  extend  beyond  the  boundary  of  Alt  one 


82.]  OF   A   SERIES   OF   FUNCTIONS  133 

00 

complete  and  distinct   part  of  the  region  of  continuity  of   2  fi(z);    and 

i  =  \ 

therefore,  by  using  the  theorem  proved  in  §  81,  it  follows  that : 

A  series  of  functions  of  a  variable,  which  converges  within  a  continuous  part 
of  the  plane  of  the  variable  z,  is  either  a  partial  or  a  complete  representation 
of  a  single  uniform,  analytic  function  of  the  variable  in  that  part  of  the  plane. 

83.  Further,  it  has  just  been  proved  that  the  converging  series  of 
functions  can,  in  any  of  the  regions  A,  be  changed  into  an  equivalent 
uniform,  analytic  function,  the  equivalence  being  valid  for  all  points  in 
that  region,  say 

2 /(•).  4(4 

i  =  l 

But  for  any  point  within  A,  the  function  Fl(z)  has  a  uniform  finite  derivative 

oo 

(§  21);  and  therefore  also    2   fi(z)  has  a  uniform  finite  derivative.     The 

i=l 

equivalence  of  the  analytic  function  and  the  series  of  functions  has  not  been 
proved  for  points  on  the  boundary;  even  if  they  are  equivalent  there,  the 
function  I\  (z)  cannot  be  proved  to  have  a  uniform  finite  derivative  at  every 

00 

point  on  the  boundary  of  A,  and  therefore  it  cannot  be  affirmed  that  2  ft  (z) 

i=\ 

has,  of  necessity,  a  uniform,  finite  derivative  at  points  on  the  boundary  of  A,  even 

oo 

though  the  value  of  2   fi(z)  be  uniform  and  finite  at  every  point  on  the 

i=l 

boundary*. 

Ex.  In  illustration  of  the  inference  just  obtained,  regarding  the  derivative  of  a 
function  at  a  point  on  the  boundary  of  its  region  of  continuity,  consider  the  series 

g(z)=  2  &V", 

n=0 

where  b  is  a  positive  quantity  less  than  unity,  and  a  is  a  positive  quantity  which  will  be 
taken  to  be  an  odd  integer. 

For  points  within  and  on  the  circumference  of  the  circle  \z  =1,  the  series  converges 
uniformly  and  unconditionally;  and  for  all  points  without  the  circle  the  series  diverges. 
It  thus  defines  a  function  for  points  within  the  circle  and  on  the  circumference,  but  not 
for  points  without  the  circle. 

Moreover  for  points  actually  within  the  circle  the  function  has  a  first  derivative  and 
consequently  has  any  number  of  derivatives.  But  it  cannot  be  declared  to  have  a 
derivative  for  points  on  the  circle:  and  it  will  in  fact  now  be  proved  that,  if  a  certain 
condition  be  satisfied,  the  derivative  for  variations  at  any  point  on  the  circle  is  not  merely 
infinite  but  that  the  sign  of  the  infinite  value  depends  upon  the  direction  of  the  variation, 
so  that  the  function  is  not  monogenic  for  the  circumference  t. 

*  It  should  be  remarked  here,  as  at  the  end  of  §  21,  that  the  result  in  itself  does  not  contravene 
Biemann's  definition  of  a  function,  according  to  which  (§  8)  -^  must  have  the  same  value  what 
ever  be  tbe  direction  of  the  vanishing  quantity  dz ;  at  a  point  on  the  boundary  of  the  region 
there  are  outward  directions  for  which  die  is  not  defined. 

t  The  following  investigation  is  due  to  Weierstrass,  who  communicated  it  to  Du  Bois-Eeymond : 
see  Crclle,  t.  Ixxix,  (1875),  pp.  29—31. 


134 


A   SERIES   OF   FUNCTIONS 


[83. 


Let  z  =  eei:  then,   as  the  function  converges  unconditionally  for  all  points  along  the 
circle,  we  take 

f(ff)=   2  lnea"ei, 

71=0 

where  6  is  a  real  variable.     Hence 


m-l  IV,an(0  +  4>)*_,,«WWl 

=  s«nH  -  —  - 
H=O          1          an$  J 

/•ea">+»>  (0  +  <f>)  i  _  ea™+«0(S 

+    2  &w  +  M  -  -T  -  1  1 

«=o  I  9  J 


assuming  m,  in  the  first  place,  to  be  any  positive  integer.     To  transform  the  first  sum  on 
the  right-hand  side,  we  take 


and  therefore 


pan  (0 + <j>)  i  _    a"0i 


2  (ab}n 

n=0 

<M21^n      8Jn(fr-*) 


if  ab>\.     Hence,  on  this  hypothesis,  we  have 

2  (ab)n  \ — \  =y    r     i  » 

*=o  (  a"0          J         ao  - 1 

where  7  is  a  complex  quantity  with  modulus  <1. 

To  transform  the  second  sum  on  the  right-hand  side,  let  the  integer  nearest  to  am 
be  am,  so  that 

7T 

for  any  value  of  m :   then  taking 

we  have  \tr^-x>  —  %n, 

and  cos  x  is  not  negative.     We  choose  the  quantity  <f>  so  that 


and  therefore 


TT  am 

ff)  —  —  —  , 
0 

which,  by  taking  m  sufficiently  large  (a  is  >  1),  can  be  made  as  small  as  we  please.     We 
now  have 

am+"(6  +<i>)i  =  Qaniti  (1  +  o™)  _  _  /  _  j  N°™ 

if  a  be  an  odd  integer,  and 

_ 


am+nOi  _  ani  (x  +  iram]  _  /  _  j  \<»meana;i 
,    a"xi 


Hence 

CD  /• 

and  therefore          2  &-  +  «  f 

,,=0  i 


_ 


-  (  -  1) 


2  6" 


83.]  MAY   NOT   POSSESS   A   DERIVATIVE  135 

The  real  part  of  the  series  on  the  right-hand  side  is 

2  bn{l  +  cosanx}; 

n=0 

every  term  of  this  is  positive  and  therefore,  as  the  first  term  is  1  +  cos  x,  the  real  part 

>  1+cos.r 

>1 
for  cos  x  is  not  negative  ;  and  it  is  finite,  for  it  is 

<2  2  bn 

K=0 

2 

<r^6- 

Moreover  far  <  TT  —  x  <  frr, 

so  that  --  is  positive  and  >-.     Hence 
TT  —  x  6 


where  TJ  is  a  finite  complex  quantity,  the  real  part  of  which  is  positive  and  greater  than 
unity.     We  thus  have 


where  |y'|<l,  and  the  real  part  of  77  is  positive  and  >  1. 
Proceeding  in  the  same  way  and  taking 

IT      '    am    ' 

TT+X 

so  that  %  =  — —  , 

we  find  — — — — t_LJ  —  _  ( _  iy™  (a^ 

where  |y/|<l  and  the  real  part  of  TJ^  a  finite  complex  quantity,  is  positive  and  greater 
than  unity. 

If  now  we  take  ab  -  1  >  fn-, 

the  real  parts  of  -  —  +  y  -*-—= ,  say  of  f, 

O  7T  (tO  —  1 

and  of  |li+yi'__L_,sayof  fl, 

are  both  positive  and  different  from  zero.     Then,  since 


and  ~x-         =  (_!)«-  (ab)m  d , 

/(. 

m  being  at  present  any  positive  integer,  we  have  the  right-hand  sides  essentially  different 
quantities,  because  the  real  part  of  the  first  is  of  sign  opposite  to  the  real  part  of  the  second. 

Now   let   m  be   indefinitely   increased;    then    $    and   x  are   infinitesimal   quantities 
which  ultimately  vanish ;  and  the  limit  of  -  [/(#  +  </>)-/(#)]  for  $  =  0  is  a  complex  infinite 


136  ANALYTICAL   EXPRESSION  [83. 

quantity  with  its  real  part  opposite  in  sign  to  the  real  part  of  the  complex  infinite  quantity 
which  is  the  limit  of  $  —      ~^     f°r     =  ®-     If#    had  a  differential  coefficient 


A 

these  two  limits  would  be  equal  :  hence  /  (0)  has  not,  for  any  value  of  6,  a  determinate 
differential  coefficient. 

From  this  result,  a  remarkable  result  relating  to  real  functions  may  be  at  once  derived. 
The  real  part  of  /  (<9)  is 

2  6ncos(an<9), 

n=0 

which  is  a  series  converging  uniformly  and  unconditionally.     The  real  parts  of 

-(-ir  («&)-<: 

and  of  +(-l)am(a6)TOf1 

are  the  corresponding  magnitudes  for  the  series  of  real  quantities  :  and  they  are  of  opposite 
signs.  Hence  for  no  value  of  6  has  the  series 

2  6"cos(an<9) 

n=0 

a  determinate  differential  coefficient,  that  is,  we  can  choose  an  increase  <£  and  a  decrease  ^ 
of  6,  both  being  made  as  small  as  we  please  and  ultimately  zero,  such  that  the  limits  of 
the  expressions 


0  -X 

are  different  from  one  another,  provided  a  be  an  odd  integer  and  ab  >  1  +|TT. 

The  chief  interest  of  the  above  investigation  lies  in  its  application  to  functions  of  real 
variables,  continuity  in  the  value  of  which  is  thus  shewn  not  necessarily  to  imply  the 
existence  of  a  determinate  differential  coefficient  defined  in  the  ordinary  way.  The 
application  is  due  to  Weierstrass,  as  has  already  been  stated.  Further  discussions  will 
be  found  in  a  paper  by  Wiener,  Crelle,  t.  xc,  (1881),  pp.  221  —  252,  in  a  remark  by 
Weierstrass,  Abh.  aus  der  Functionenlehre,  (1886),  p.  100,  and  in  a  paper  by  Lerch,  Crelle, 
t.  ciii,  (1888),  pp.  126  —  138,  who  constructs  other  examples  of  continuous  functions  of 
real  variables  ;  and  an  example  of  a  continuous  function  without  a  derivative  is  given  by 
Schwarz,  Ges.  Werke,  t.  ii,  pp.  269  —  274. 

The  simplest  classes  of  ordinary  functions  are  characterised  by  the  properties  :  — 
(i)     Within  some  region  of  the  plane  of  the  variable  they  are  uniform,  finite  and 

continuous  : 
(ii)    At  all  points  within  that  region  (but  not  necessarily  on  its  boundary)  they  have 

a  differential  coefficient  : 

(iii)   When  the  variable  is  real,  the  number  of  maximum  values  and  the  number  of 
minimum  values  within  any  given  range  is  finite. 

The  function     2  bn  cos  (anQ\  suggested  by  Weierstrass,  possesses  the  first  but  not  the 

71=0 

second  of  these  properties.  Kb'pcke  (Math.  Ann.,  t.  xxix,  pp.  123  —  140)  gives  an  example 
of  a  function  which  possesses  the  first  and  the  second  but  not  the  third  of  these 
properties. 

84.  In  each  of  the  distinct  portions  Alt  A.2>...  of  the  complete  region  of 
continuity  of  a  series  of  functions,  the  series  can  be  represented  by  a 
monogenic  analytic  function,  the  elements  of  which  are  converging  power- 
series.  But  the  equivalence  of  the  function  -series  and  the  monogenic 


84.]  REPRESENTING   DIFFERENT   FUNCTIONS  137 

analytic  function  for  any  portion  A^  is  limited  to  that  region.  When  the 
monogenic  analytic  function  can  be  continued  from  A^  into  Az,  the  continua 
tion  is  not  necessarily  the  same  as  the  monogenic  analytic  function  which  is 

00 

the  equivalent  of  the  series  2  fi(z)  in  A2.     Hence,  if  the  monogenic  analytic 

i  =  l 

functions  for  the  two  portions  A^  and  A2  be  different,  the  function-series 
represents  different  functions  in  the  distinct  parts  of  its  region  of  continuity. 

A  simple  example  will  be  an  effective  indication  of  the  actual  existence 
of  such  variety  of  representation  in  particular  cases  ;  that,  which  follows,  is 
due  to  Tannery*. 

Let  a,  b,  c  be  any  three  constants ;  then  the  fraction 

a  +  bczm 
Y+'bzm  ' 
when  m  is  infinite,  is  equal  to  a  if  z  \  <  1,  and  is  equal  to  c  if  |  z  >  1. 

Let  m0,  m1}  m2>...  be  any  set  of  positive  integers  arranged  in  ascending 
order  and  be  such  that  the  limit  of  mn,  when  n  =  oo ,  is  infinite.  Then, 

since 

a  +  bczm*     a  +  bczm°       »    {a  +  bczmi     a  +  bczm 


1  +  bzm»        1  -f  bzm°      f.i  (1  +  bzmi       I  +  bz'" 
^mo 

"  ~*  a) 


the  function  <f)(z),  defined  by  the  equation 

,.      a  +  bczm°      .,         N  S    f    0^-^-1-1)^-1 

+  (z}  =  TT6^  +  b  (G  ~  a)  £  {(I  +  bz^)  (i  +  6^ 

converges  uniformly  and  unconditionally  to  a  value  a  if  \  z  <  1,  awe?  converges 
uniformly  and  unconditionally  to  a  value  c  if  z  \  >  1.  But  it  does  not  con 
verge  uniformly  and  unconditionally  if  z  \  =  1. 

The  simplest  case  occurs  when  b  =  —  1  and  m^  =  2*  ;  then,  denoting  the 
function  by  <f>  (z),  we  have 


a  -  cz     ,         .  (    z  z2  z4 


that  is,  the  function  <f>  (z)  is  equal  to  a  if    z  <  1,  and  it  is  equal  to  c  if 


*  It  is  contained  in  a  letter  of  Tannery's  to  Weierstrass,  who  communicated  it  to  the  Berlin 
Academy  in  1881,  Abh.  aus  der  Functionenlehre,  pp.  103,  104.  A  similar  series,  which  indeed  is 
equivalent  to  the  special  form  of  $  (z),  was  given  by  Schroder,  Schlfim.  Zeitschrift,  t.  xxii,  (1876), 
p.  184;  and  Pringsheim,  Math.  Ann.,  t.  xxii,  (1883),  p.  110,  remarks  that  it  can  be  deduced, 
without  material  modifications,  from  an  expression  given  by  Seidel,  Crelle,  t.  Ixxiii,  (1871), 
pp.  297-  -299. 


138  LINE   OF   SINGULARITIES  [84. 

When  \z  =\,  the  function  can  have  any  value  whatever.  Hence  a  circle 
of  radius  unity  is  a  line  of  singularities,  that  is,  it  is  a  line  of  discontinuity 
for  the  series.  The  circle  evidently  has  the  property  of  dividing  the  plane 
into  two  parts  such  that  the  analytical  expression  represents  different 
functions  in  the  two  parts. 

If  we  introduce  a  new  variable  £  connected  with  z  by  the  relation* 

l  +z 


then,  if  £=  £  +  iy  and  z  =  x  +  iy,  we  have 

1     rfS.    nil 

fc       i  —  x      y 


so  that  £  is  positive  when  \z\<  1,  and  £  is  negative  when  \  z  \  >  1.     If  then 


the  function  %(£)  is  equal  to  a  or  to  c  according  as  the  real  part  of  f  is 
positive  or  negative. 

And,  generally,  if  we  take  £  a  rational  function  of  z  and  denote  the 
modified  form  of  </>  (£),  which  will  be  a  sum  of  rational  functions  of  z,  by 
^(z),  then  <f>i(z)  will  be  equal  to  a  in  some  parts  of  the  plane  and  to  c 
in  other  parts  of  the  plane.  The  boundaries  between  these  parts  are  lines 
of  singular  points  :  and  they  are  constituted  by  the  ^-curves  which  correspond 
to  £|  =  1. 

85.  Now  let  F(z)  and  G(z)  be  two  functions  of  z  with  any  number  of 
singularities  in  the  plane  :  it  is  possible  to  construct  a  function  which  shall 
be  equal  to  F  (z}  within  a  circle  centre  the  origin  and  to  G  (z)  without  the 
circle,  the  circumference  being  a  line  of  singularities.  For,  when  we  make 
a  =  1  and  c  =  0  in  </>  (z)  of  §  84,  the  function 

1  z  z*  z4 

00)=-  --  +  -.  —  r  +  •-:   —  :   +  -.  -^r  +  .  .  . 
V/        1—0        Z2  —  I        Z*—  I        ZS  —I 

is  unity  for  all  points  within  the  circle  and  is  zero  for  all  points  without  it  : 

and  therefore 

G(z}  +  {F(z)-G(z)}6(z} 

is  a  function  which  has  the  required  property. 

Similarly  F3  (z)  +  {F,  (z)  -  F,  (z)}  6  (z)  +  {F,  (z)  -  F3  (z}}  6  ( 


is  a  function  which  has  the  value  Fl  (z)  within  a  circle  of  radius  unity,  the  value  F2  (z) 
between  a  circle  of  radius  unity  and  a  concentric  circle  of  radius  r  greater  than  unity,  and 
the  value  F3(z)  without  the  latter  circle.  All  the  singularities  of  the  functions  F1}  F2,  F3 
are  singularities  of  the  function  thus  represented;  and  it  has,  in  addition  to  these,  the 
two  lines  of  singularities  given  by  the  circles. 

*  The  significance  of  a  relation  of  this  form  will  be  discussed  in  Chapter  XIX. 


85.]  MONOGENIC   FUNCTIONALITY  139 


Again,  6 

is  a  function  of  s,  which  is  equal  to  F(z)  on  the  positive  side  of  the  axis  of  y,  and  is  equal 
to  G  (z)  on  the  negative  side  of  that  axis. 

1+2 

Also,  if  we  take  £e      l  —p\  =  ^~i 

where  ax  and  p1  are  real  constants,  as  an  equation  defining  a  new  variable  £  +  iy,  we  have 
|  cos  at  +  77  sin  aj  -pl  =  p. \23T~2 

so  that  the  two  regions  of  the  2-plane  determined  by  \z\<l  and  \z\>l  correspond  to  the 
two  regions  of  the  {"-plane  into  which  the  line  £  cos  a:  +  77  sin  al—p1  =  0  divides  it.     Let 

,-«'ai  —  »,  — 1\ 


so  that  on  the  positive  side  of  the  line  £  cos  at  +  77  sin  aj  —  p1  =  0  the  function  6l  is  unity  and 
on  the  negative  side  of  that  line  it  is  zero.  Take  any  three  lines  defined  by  ax,  p1;  a2,  p2', 
a,,  pn  respectively  ;  then 

AJ.A11  (2)\-F/(l) 


is  a  function  which  has  the  value  F  within 
the  triangle,  the  value  -  F  in  three  of  the 
spaces  without  it,  and  the  value  zero  in  the 
remaining  three  spaces  without  it,  as  indi 
cated  in  the  figure  (fig.  13). 

And  for  every  division  of  the  plane  by 
lines,  into  which  a  circle  can  be  transformed    (3) 
by  rational  equations,  as  will  be  explained 
when  conformal  representation  is  discussed  (1)  / 

hereafter,  there  is  a  possibility  of  represent-  Fig.  13. 

ing  discontinuous  functions,  by  expressions  similar  to  those  just  given. 

These  examples  are  sufficient  to  lead  to  the  following  result*,  which  is 
complementary  to  the  theorem  of  §  82  : 

When  the  region  of  continuity  of  an  infinite  series  of  functions  consists 
of  several  distinct  parts,  the  series  represents  a  single  function  in  each  part 
but  it  does  not  necessarily  represent  the  same  function  in  different  parts. 

It  thus  appears  that  an  analytical  expression  of  given  form,  which  con 
verges  uniformly  and  unconditionally  in  different  parts  of  the  plane  separated 
from  one  another,  can  represent  different  functions  of  the  variable  in  those 
different  parts ;  and  hence  the  idea  of  monogenic  functionality  of  a  complex 
variable  is  not  coextensive  with  the  idea  of  functional  dependence  expressible 
through  arithmetical  operations,  a  distinction  first  established  by  Weierstrass. 

86.  We  have  seen  that  an  analytic  function  has  not  a  definite  value  at 
an  essential  singularity  and  that,  therefore,  every  essential  singularity  is 
excluded  from  the  region  of  definition  of  the  function. 

*  Weierstrass,  I.e.,  p.  90. 


140  SINGULAR  LINES  [86. 

Again,  it  has  appeared  that  not  merely  must  single  points  be  on  occasion 
excluded  from  the  region  of  definition  but  also  that  functions  exist  with 
continuous  lines  of  essential  singularities  which  must  therefore  be  excluded. 
One  method  for  the  construction  of  such  functions  has  just  been  indicated  : 
but  it  is  possible  to  obtain  other  analytical  expressions  for  functions  which 
possess  what  may  be  called  a  singular  line.  Thus  let  a  function  have  a 
circle  of  radius  c  as  a  line  of  essential  singularity*;  let  it  have  no  other 
singularities  in  the  plane  and  let  its  zeros  be  al}  a2,  a3,...,  supposed  arranged 
in  such  order  that,  if  pneie"  =  an>  then 

I  Pn       C  |  ^    Pn+i  ~  C   > 

so  that  the  limit  of  pn,  when  n  is  infinite,  is  c. 

Let  cn  =  ceie«,  a  point  on  the  singular  circle,  corresponding  to  an  which  is 
assumed  not  to  lie  on  it.  Then,  proceeding  as  in  Weierstrass's  theory  in  §  51, 
if 

«.=  oo    („  _ 

Gz=  n 


where        gn(z)  =      -     +         L_        +...  +  _ 

Z-Cn         2  \Z-CnJ  mn  -  I  \  Z  -  Cn 

G  (z)  is  a  uniform  function,  continuous  everywhere  in  the  plane  except  along 
the  circumference  of  the  circle  which  may  be  a  line  of  essential  singularities. 

Special  simpler  forms  can  be  derived  according  to  the  character  of  the 
series  of  quantities  constituted  by  |  an  -  cn  .     If  there  be  a  finite  integer  m, 

00 

such  that    2    an  —  cn  m  is  a  converging  series,  then  in  gn  (z)  only  the  first 

M  =  l 

m  —  1  terms  need  be  retained. 
Ex.     Construct  the  function  when 


m  being  a  given  positive  integer  and  r  a  positive  quantity. 

Again,  the  point  cn  was  associated  with  an  so  that  they  have  the  same 
argument :  but  this  distribution  of  points  on  the  circle  is  not  necessary  and 
can  be  made  in  any  manner  which  satisfies  the  condition  that  in  the  limited 

00 

case  just  quoted  the  series  2    an  —  cn  m  is  a  converging  series. 

Singular  lines  of  other  classes,  for  example,  sectioiis\  in  connection  with  functions 
defined  by  integrals,  arise  in  connection  with  analytical  functions.  They  are  discussed 
by  Painleve,  "Sur  les  lignes  singulieres  des  fonctions  analytiques,"  (These,  Gauthier- 
Villars,  Paris,  1887). 

Ex.     Shew  that,  if  the  zeros  of  a  function  be  the  points 

.  _b+c—  (a  —  d)  i 

ZT  ^    ~7  i    7T          \    •  5 


*  This  investigation  is  due  to  Picard,  Comptes  Rendus,  t.  xci,  (1881),  pp.  690—692. 
t  Called  conpures  by  Hermite  ;  see  §  103. 


86.]  LACUNARY    FUNCTIONS  141 

where  a,  ?;,  c,  d  are   integers  satisfying  the  condition   ad-bo  =  l,  so  that  the  function 
has  a  circle  of  radius  unity  for  an  essential  singular  line,  then  if 

b  +  di 


„ 

2J  =  -^  -  =—  ,  , 

d+bi' 


(        A 

the  function  n  \ 5  e  z 

(z  —  li 

where  the  product  extends  to  all  positive  integers  subject  to  the  foregoing  condition 
ad-bc  =  l,  is  a  uniform  function  finite  for  all  points  in  the  plane  not  lying  on  the 
circle  of  radius  unity.  (Picard.) 

87.  In  the  earlier  examples,  instances  were  given  of  functions  which 
have  only  isolated  points  for  their  essential  singularities :  and,  in  the  later 
examples,  instances  have  been  given  of  functions  which  have  lines  of 
essential  singularities,  that  is,  there  are  continuous  lines  for  which  the 
functions  do  not  exist.  We  now  proceed  to  shew  how  functions  can  be 
constructed  which  do  not  exist  in  assigned  continuous  spaces  in  the  plane, 
these  spaces  being  aggregates  of  essential  singularities.  Weierstrass  was 
the  first  to  draw  attention  to  lacunary  functions,  as  they  may  be  called ; 
the  following  investigation  in  illustration  of  Weierstrass's  theorem  is  due  to 
Poincare'  *. 

Take  any  convex  curve  in  the  plane,  say  G ;  and  consider  the  function 

*z^b' 
where  the  quantities  A  are  constants,  subject  to  the  conditions 

(i)     The  series  ^\A\  converges  uniformly  and  unconditionally : 
(ii)     Each  of  the  points  b  is  either  within  or  on  the  curve  G : 
(iii)     The  points  b  are  the  aggregate  of  all  rational  j  points  within  and 
on  C :    then  the   function   is  a   uniform    analytical    function   for   all    points 
without  C  and  it  has  the  area  of  G  for  a  lacunary  space. 

First,  it  is  evident  that,  if  z  =  b,  then  the  series  does  not  converge. 
Moreover  as  the  points  b  are  the  aggregate  of  all  the  rational  points  within 
or  on  C,  there  will  be  an  infinite  number  of  singularities  in  the  immediate 
vicinity  of  b :  we  shall  thus  have  an  unlimited  number  of  terms  each  infinite 
of  the  first  order,  and  thus  (§  42)  the  point  b  will  be  an  essential  singularity. 
As  this  is  true  of  all  points  z  within  or  on  C,  it  follows  that  the  area  C  is  a 
lacunary  space  for  the  function,  if  the  function  exist  at  all. 

Secondly,  let  z  be  a  point  without  G ;  and  let  d  be  the  distance  of  z  from 
the  nearest  point  of  the  boundary  of  C^f%  so  that  d  is  not  a  vanishing  quantity. 

*  Acta  Soc.  Fenn.,  t.  xii,  (1883),  pp.  341—350. 

J  Rational  points  within  or  on  C  are  points  whose  positions  can  be  determined  rationally  in 
terms  of  the  coordinates  of  assigned  points  on  C ;  examples  will  be  given. 

t  This  will  be  either  the  shortest  normal  from  z  to  the  boundary  or  the  distance  of  z  from 
some  point  of  abrupt  change  of  direction,  as  for  instance  at  the  angular  point  of  a  polygon. 


142 


FUNCTIONS   WITH 


[87. 


Then  |  z  —  b  \  ^  d ;  and  therefore 

A      _    \A\       \A\ 
~\z-b\<  d  ' 


z-b 


so  that 


-b 


A 


z-b 


Now  2  j.A|  converges  uniformly  and  unconditionally  and  therefore,  as  d  does 
not  vanish, 


z-b 
converges  uniformly  and  unconditionally,  that  is. 


is  a  function  of  2  which  converges  uniformly  and  unconditionally  for  every 
point  without  C.  Let  it  be  denoted  by  <£  (z). 

Let  c  be  any  point  without  C,  and  let  r  be  the  radius  of  the  greatest 
circle  centre  c  which  can  be  drawn  so  as  to  have  no  point  of  C  within  itself 
or  on  its  circumference,  so  that  r  is  the  radius  of  the  domain  of  c;  then 
b  —  c  >  r,  for  all  points  b. 

If  we  take  a  point  z  within  this  circle,  we  have  \z  —  c  =6r,  where  6  <  1. 

Now  for  all  points  within  this  circle  the  function  <£  {z}  converges  uniformly, 

A 

and  every  term  --  =•  of  <f>  (z)  is  finite.     Also,  for  points  within  the  circle,  we 

A 

can  expand  --  j  in  powers  of  z  —  c  in  the  form 


of  a  converging  series.     Hence,  by  §  82,  we  have 

<£(*)=  2  Bm(z-c)m, 


a  series  converging  uniformly  and  unconditionally  for  all  points  within  the 
circle  centre  c  and  radius  r,  which  circle  is  the  circle  of  convergence  of  the 
series.  The  function  can  be  expressed  in  the  usual  manner  over  the  whole  of 
the  region  of  continuity,  which  is  the  part  of  the  plane  without  the  curve  C. 

Thus  0  (z)  is  a  uniform  analytical  function,  having  the  area  of  C  for  a 
lacunary  space. 

As   an   example,  take  a  convex   polygon   having  o1}  ......  ,  ap  for  its  angular  points; 

then  any  point 

......  +mj>ap 


TOI  +  ......  +mp 

where  mlt  ......  ,  mp  are  positive  integers  or  zero  (simultaneous  zeros  being  excluded),  is 


87.]  LACUNARY    SPACES  143 

either  within  the  polygon  or  on  its  boundary  :  and  any  rational  point  within  the  polygon 

or  on  its  boundary  can  be  represented  by 

p 
2  mrar 

r=l 

P          ' 
2  mr 
r=l 
by  proper  choice  of  ?n15  ......  ,  mp,  a  choice  which  can  be  made  in  an  infinite  number  of  ways. 

Let  ult  ......  ,  Up  be  given  quantities,  the  modulus  of  each  of  which  is  less  than  unity: 

then  the  series 

•9-11  m>         11  mf 

«&  ^  '  I         ......  ftp 

o 

converges  uniformly  and  unconditionally.     Then  all  the  assigned  conditions  are  satisfied 
for  the  function 


_  ..  .  +  mpap  >  ' 

ml  +  ......  +mp    J 

and  therefore  it  is  a  function  which  converges  uniformly  and  unconditionally  everywhere 
outside  the  polygon  and  which  has  the  polygonal  space  (including  the  boundary)  for 
a  lacunary  space. 

If,  in  particular,  p  =  Z,  we  obtain  a  function  which  has  the  straight  line 
joining  ax  and  a2  as  a  line  of  essential  singularity.  When  we  take  at  =  0, 
a.2  =  1  and  slightly  modify  the  summation,  we  obtain  the  function 


2    2  ^    2 

w=l  m=0         W& 

7i 

which,  when    u^  <\   and  |w2|<l,  converges   uniformly  and  unconditionally 
everywhere  in  the  plane  except  at  points  between  0  and  1  on  the  axis  of  real 
quantities,  this  part  of  the  axis  being  a  line  of  essential  singularity. 
For  the  general  case,  the  following  remarks  may  be  made : 

(i)  The  quantities  u1}  u2>...  need  not  be  the  same  for  every  term;  a 
numerator,  quite  different  in  form,  might  be  chosen,  such  as 
(mj2+  ...  +  m/)"'1  where  2//,  > p  ;  all  that  is  requisite  is  that  the 
series,  made  up  of  the  numerators,  should  converge  uniformly 
and  unconditionally. 

(ii)  The  preceding  is  only  a  particular  illustration  and  is  not  necessarily 
the  most  general  form  of  function  having  the  assigned  lacunary 
space. 

It  is  evident  that  the  first  step  in  the  construction  of  a  function,  which 
shall  have  any  assigned  lacunary  space,  is  the  formation  of  some  expression 
which,  by  the  variation  of  the  constants  it  contains,  can  be  made  to 
represent  indefinitely  nearly  any  point  within  or  on  the  contour  of  the 
space.  Thus  for  the  space  between  two  concentric  circles  of  radii  a  and  c 
and  centre  the  origin  we  should  take 

Wja  +  O-WjU    ^a« 
-a£-  e  n 

n 


144  EXAMPLES  [87. 

which,  by  giving  m^  all  values  from  0  to  n,  ra2  all  values  from  0  to  n  —  1  and 
n  all  values  from  1  to  infinity  will  represent  all  rational  points  in  the  space : 
and  a  function,  having  the  space  between  the  circles  as  lacunary,  would  be 
given  by 


oo        n      n-1 

2    2     2 

n=l  »»!=(>  m2=0 


(n  —  raj)  b    ^  271- 


•r  /3 

.6    —  C 

n 


provided   u\  <  1,  u^  <  1,   u2  <  1. 

In  particular,  if  a  =  6,  then  the  common  circumference  is  a  line  of  essential  singularity 
for  the  corresponding  function.     It  is  easy  to  see  that  the  function 


z  —  ae  n 


ao    2n-l     m         n 

provided  the  series  2     2   u       v 

n=l  m=0      m,n    m,  n 

converges  uniformly  and  unconditionally,  is  a  function  having  the  circle  |0|  =  a  as  a  line  of 
essential  singularity. 

Other  examples  will  be  found  in  memoirs  by  Goursat*,  Poincaref,  and  HomenJ. 

Ex.  1.     Shew  that  the  function 


where  r  is  a  real  positive  quantity  and  the  summation  is  for  all  integers  m  and  n  between 
the  positive  and  the  negative  infinities,  is  a  uniform  function  in  all  parts  of  the  plane 
except  the  axis  of  real  quantities  which  is  a  line  of  essential  singularity. 

Ex.  2.     Discuss  the  region  in  which  the  function 


w=i    m=i  jf/=i             i  ^-     .     ••-     • 
2—1 1 i 

\7i          71 

is  definite.     (Homen.) 

Ex.  3.     Prove  that  the  function 

n=0 

exists  only  within  a  circle  of  radius  unity  and  centre  the  origin.  (Poincare.) 

Ex.  4.     An  infinite  number  of  points  at,  a2,  as, are  taken  on  the  circumference  of 

a  given  circle,  centre  the  origin,  so  that  they  form  the  aggregate  of  rational  points  on  the 
circumference.     Shew  that  the  series 

2    l       Z 

can  be  expanded  in  a  series  of  ascending  powers  of  z  which  converges  for  points  within  the 
circle,  but  that  the  function  cannot  be  continued  across  the  circumference  of  the  circle. 

(Stieltjes.) 

*  Comptes  Rendus,  t.  xciv,  (1882),  pp.  715—718 ;   Bulletin  de  Darboux,  2me  Ser. ,  t.  xi,  (1887), 
pp.  109—114. 

t  In  the  memoir,  quoted  p.  138,  and  Comptes  Rendus,  t.  xcvi,  (1883),  pp.  1134-1136. 
+  Acta  Soc.  Fenn.,  t.  xii,  (1883),  pp.  445—464. 


87.]  EXAMPLES  145 

Ex.  5.     Prove  that  the  series 


2  |  : 

7T    .00    -"     K1-2TO- 
9      oo 


22 


~li)2)  ' 


7T  _oo  _oo  (^(1 — 2wi — 2nz     i)  \zm-\-Anz~ 

where  the  summation  extends  over  all  positive  and  negative  integral  values  of  ra  and  of  n 
except  simultaneous  zeros,  is  a  function  which  converges  uniformly  and  unconditionally 
for  all  points  in  the  finite  part  of  plane  which  do  not  lie  on  the  axis  of  y ;  and  that 
it  has  the  value  +1  or  -  1  according  as  the  real  part  of  z  is  positive  or  negative. 

(Weierstrass.) 
Ex.  6.     Prove  that  the  region  of  continuity  of  the  series 


consists  of  two  parts,  separated  by  the  circle  z\  =  l  which  is  a  line  of  infinities  for 
the  series :  and  that,  in  these  two  parts  of  the  plane,  it  represents  two  different 
functions. 

_<a'ir 

If  two  complex  quantities  a>  and  to'  be  taken,  such  that  z  =  e  ^  and  the  real  part  of 
^.  is  positive,  and  if  they  be  associated  with  the  elliptic  function  $  (u)  as  its  half-periods, 
then  for  values  of  z  which  lie  within  the  circle  z  =  \ 


in  the  usual  notation  of  Weierstrass's  theory  of  elliptic  functions. 

Find  the  function  which  the  series  represents  for  values  of  z  without  the  circle  \z\  =  \. 

(Weierstrass.) 

Ex.  7.     Four  circles  are  drawn  each  of  radius  -^  having  their  centres  at  the  points 

1,  i,   -  1,  -i  respectively;  the  two  parts  of  the  plane,  excluded  by  the  four  circumferences, 
are  denoted  the  interior  and  the  exterior  parts.     Shew  that  the  function 

n='K  sini^TT  (     1  1  1  1 


is  equal  to  IT  in  the  interior  part  and  is  zero  in  the  exterior  part.  (Appell.) 

Ex.  8.     Obtain  the  values  of  the  function 

»;-l-  (-!)•(,  i  >1    l 

«=i         n         V1  •>      (2  +  l)«     (2-l)«J 

in  the  two  parts   of  the  area  within   a  circle  centre  the   origin  and  radius  2  which  lie 

without  two   circles   of    radius  unity,   having  their  centres  at   the  points    1    and    -  1 

respectively.  (Appell.) 


Ex-  9-     If 
and 


,~3     ...... 

amr        (2-«m)3  J 

where  the  regions  of  continuity  of  the  functions  F  extend  over  the  whole  plane,  then  /  (z) 
is  a  function  existing  everywhere  except  within  the  circles  of  radius  unity  described  round 
the  points  a,  ,  «2,  ......  ,  an.  (Teixeira.) 

F-  10 


146  CLASSIFICATION  [87. 

Ex.   10.     Let  there   be  n  circles  having  the  origin  for  a  common  centre,  and  let 
£,,  (72,  ......  ,  (7n,  C'n  +  1  be  %  +  1  arbitrary  constants;  also  let  a1}  a2,  ......  ,  an  be  any  w  points 

lying  respectively  on  the   circumferences  of  the  first,  the   second,  ......  ,  the  nth   circles. 

Shew  that  the  expression 


1  ("(CL 
27T./0    W* 


has  the  value  <7m  for  points  z  lying  between  the  (»w  -  l)th  and  the  with  circles  and  the 
value  (7n  +  1  for  points  lying  without  the  nth  circle. 

Construct  a  function  which  shall  have  any  assigned  values  in  the  various  bands  into 
which  the  plane  is  divided  by  the  circles.  (Pincherle.) 

88.  In  §  32  it  was  remarked  that  the  discrimination  of  the  various 
species  of  essential  singularities  could  be  effected  by  means  of  the  properties 
of  the  function  in  the  immediate  vicinity  of  the  point. 

Now  it  was  proved,  in  §  63,  that  in  the  vicinity  of  an  isolated  essential 
singularity  b  the  function  could  be  represented  by  an  expression  of  the  form 


for  all  points  in  the  space  without  a  circle  centre  b  of  small  radius  and  within 
a  concentric  circle  of  radius  not  large  enough  to  include  singularities  at 
a  finite  distance  from  b.  Because  the  essential  singularity  at  b  is  isolated, 
the  radius  of  the  inner  circle  can  be  diminished  to  be  all  but  infinitesimal : 

the  series  P  (z  —  b)  is  then  unimportant  compared  with  G  I  —31 ) ,  which 
can  be  regarded  as  characteristic  for  the  singularity  of  the  function. 

Another  method  of  obtaining  a  function,  which  is  characteristic  of  the 
singularity,  is  provided  by  §  68.  It  was  there  proved  that,  in  the  vicinity  of 
an  essential  singularity  a,  the  function  could  be  represented  by  an  expression 
of  the  form 


where,  within  a  circle  of  centre  a  and  radius  not  sufficiently  large  to  include 
the  nearest  singularity  at  a  finite  distance  from  a,  the  function  Q  (z  —  a)  is 
finite  and  has  no  zeros :  all  the  zeros  of  the  given  function  within  this  circle 
(except  such  as  are  absorbed  into  the  essential  singularity  at  a)  are  zeros  of 

the  factor  H  (  -  -  ] ,  and  the  integer-index  n  is  affected  by  the  number  of  these 
zeros.  When  the  circle  is  made  small,  the  function 


z-a 


can  be  regarded  as  characteristic  of  the  immediate  vicinity  of  a  or,  more 
briefly,  as  characteristic  of  a. 


88.]  OF   SINGULARITIES  147 

It  is  easily  seen  that  the  two  characteristic  functions  are  distinct.  For 
if  F  and  F^  be  two  functions,  which  have  essential  singularities  at  a  of  the 
same  kind  as  determined  by  the  first  characteristic,  then 

F(z)-Fl(z)  =  P(z-a)-Pl(z-a) 
=  P,(z-a\ 

while  if  their  singularities  at  a  be  of  the  same  kind  as  determined  by  the 
second  characteristic,  then 

F(z)_Q(z-a) 
f\(*)-Q^-~a)  =  Q^2- 

in  the  immediate  vicinity  of  a,  since  Q1  has  no  zeros.     Two  such  equations 
cannot  subsist  simultaneously,  except  in  one  instance. 

Without  entering  into  detailed  discussion,  the  results  obtained  in  the 
preceding  chapters  are  sufficient  to  lead  to  an  indication  of  the  classification 
of  singularities*. 

Singularities  are  said  to  be  of  the  first  class  when  they  are  accidental ; 
and  a  function  is  said  to  be  of  the  first  class  when  all  its  singularities  are  of 
the  first  class.  It  can,  by  §  48,  have  only  a  finite  number  of  such  singularities, 
each  singularity  being  isolated. 

It  is  for  this  case  alone  that  the  two  characteristic  functions  are  in 
accord. 

When  a  function,  otherwise  of  the  first  class,  fails  to  satisfy  the  last 
condition,  solely  owing  to  failure  of  finiteness  of  multiplicity  at  some  point, 
say  at  z  =  x  ,  then  that  point  ceases  to  be  an  accidental  singularity.  It  has 
been  called  (§  32)  an  essential  singularity  ;  it  belongs  to  the  simplest  kind  of 
essential  singularity ;  and  it  is  called  a  singularity  of  the  second  class. 

A  function  is  said  to  be  of  the  second  class  when  it  has  some  singularities 
of  the  second  class  ;  it  may  possess  singularities  of  the  first  class.  By  an 
argument  similar  to  that  adopted  in  §  48,  a  function  of  the  second  class 
can  have  only  a  limited  number  of  singularities  of  the  second  class,  each 
singularity  being  isolated. 

When  a  function,  otherwise  of  the  second  class,  fails  to  satisfy  the  last 
condition  solely  owing  to  unlimited  condensation  at  some  point,  say  at  z  =  oo  , 
of  singularities  of  the  second  class,  that  point  ceases  to  be  a  singularity 
of  the  second  class:  it  is  called  a  singularity  (necessarily  essential)  of  the 
third  class. 

*  For  a  detailed  discussion,  reference  should  be  made  to  Guichard,  "  Theorie  des  points 
singnhers  essentiels"  (These,  Gauthier-ViUars,  Paris,  1883),  who  gives  adequate  references  to  the 

:stigations  of  Mittag-Leffler  in  the  introduction  of  the  classification  and  to  the  researches  of 
Cantor.  See  also  Mittag-Leffler,  Acta  Math.,  t.  iv,  (1884),  pp.  1_79;  Cantor  Crelle  t  Ixxxiv 
1878),  pp.  242—258,  Acta  Math.,  t.  ii,  (1883),  pp.  311—328. 

10—2 


148  CLASSIFICATION    OF    SINGULARITIES  [88. 

A  function  is  said  to  be  of  the  third  class  when  it  has  some  singularities 
of  the  third  class ;  it  may  possess  singularities  of  the  first  and  the  second 
classes.  But  it  can  have  only  a  limited  number  of  singularities  of  the  third 
class,  each  singularity  being  isolated. 

Proceeding  in  this  gradual  sequence,  we  obtain  an  unlimited  number  of 
classes  of  singularities:  and  functions  of  the  various  classes  can  be  constructed 
by  means  of  the  theorems  which  have  been  proved.  A  function  of  class  n 
has  a  limited  number  of  singularities  of  class  n,  each  singularity  being 
isolated,  and  any  number  of  singularities  of  lower  classes  which,  except  in  so 
far  as  they  are  absorbed  in  the  singularities  of  class  n,  are  isolated  points. 

The  effective  limit  of  this  sequence  of  classes  is  attained  when  the 
number  of  the  class  increases  beyond  any  integer,  however  large.  When 
once  such  a  limit  is  attained,  we  have  functions  with  essential  singularities  of 
unlimited  class,  each  singularity  being  isolated ;  when  we  pass  to  functions 
which  have  their  essential  singularities  no  longer  isolated  but,  as  in  previous 
class-developments,  of  infinite  condensation,  it  is  necessary  to  add  to  the 
arrangement  in  classes  an  arrangement  in  a  wider  group,  say,  in  species*. 

Calling,  then,  all  the  preceding  classes  of  functions  functions  of  the  first 
species,  we  may,  after  Guichard  (I.e.),  construct,  by  the  theorems  already 
proved,  a  function  which  has  at  the  points  al}  a*,...  singularities  of  classes 
1,  2,...,  both  series  being  continued  to  infinity.  Such  a  function  is  called 
a  function  of  the  second  species. 

By  a  combination  of  classes  in  species,  this  arrangement  can  be  continued 
indefinitely ;  each  species  will  contain  an  infinitely  increasing  number  of 
classes;  and  when  an  unlimited  number  of  species  is  ultimately  obtained, 
another  wider  group  must  be  introduced. 

This  gradual  construction,  relative  to  essential  singularities,  can  be  carried 
out  without  limit  ;  the  singularities  are  the  characteristics  of  the  functions. 

*  Guichard  (I.e.)  uses  the  term  genre. 


CHAPTER   VIII. 

MULTIFORM  FUNCTIONS. 

89.  HAVING  now  discussed  some  of  the  more  important  general  properties 
of  uniform  functions,  we  proceed  to  discuss  some  of  the  properties  of  multiform 
functions. 

Deviations  from  uniformity  in  character  may  arise  through  various  causes : 
the  most  common  is  the  existence  of  those  points  in  the  ^-plane,  which  have 
already  (§  12)  been  defined  as  branch-points. 

As  an  example,  consider  the  two  power-series 

Wl  =  l-i/-i/2-...     ,         W2  =  _(i_i/_^_...  )f 

which,  for  points  in  the  plane  such  that  z'  is  less  than  unity,  are  the  two 
values  of  (1  -  /)* ;  they  may  be  regarded  as  two  branches  of  the  function  w 
defined  by  the  equation 

w2  =  1  —  z'  =  z. 

Let  /  describe  a  small  curve  (say  a  circle  of  radius  r)  round  the  point 
z'  =  l,  beginning  on  the  axis  of  x\  the  point  1  is  the  origin  for  z.  Then  z 
is  r  initially,  and  at  the  end  of  the  first  description  of  the  circle  z  is  re2wi ; 
hence  initially  wl  is  + 14  and  w.2  is  -  r*}  and  at  the  end  of  the  description 
w1  is  -f  r^e™  and  w2  is  —  r^e™,  that  is,  wl  is  —  rf  and  w.2  is  +  ri  Thus  the 
effect  of  the  single  circuit  is  to  change  wl  into  w.2  and  w2  into  w1}  that  is, 
the  effect  of  a  circuit  round  the  point,  at  which  w1  and  w2  coincide  in  value, 
is  to  interchange  the  values  of  the  two  branches. 

If,  however,  z  describe  a  circuit  which  does  not  include  the  branch-point, 
wl  and  w2  return  each  to  its  initial  value. 

Instances  have  already  occurred,  e.g.  integrals  of  uniform  functions,  in 
which  a  variation  in  the  path  of  the  variable  has  made  a  difference  in  the 


150  CONTINUATIONS  [89. 

result;  but  this  interchange  of  value  is  distinct  from  any  of  the  effects 
produced  by  points  belonging  to  the  families  of  critical  points  which  have 
been  considered.  The  critical  point  is  of  a  new  nature ;  it  is,  in  fact,  a 
characteristic  of  multiform  functions  at  certain  associated  points. 

We  now  proceed  to  indicate  more  generally  the  character  of  the  relation 
of  such  points  to  functions  affected  by  them. 

The  method  of  constructing  a  monogenic  analytic  function,  described  in 
§  34,  by  forming  all  the  continuations  of  a  power-series,  regarded  as  a  given 
initial  element  of  the  function,  leads  to  the  aggregate  of  the  elements  of  the 
function  and  determines  its  region  of  continuity.  When  the  process  of  con 
tinuation  has  been  completely  carried  out,  two  distinct  cases  may  occur. 

In  the  first  case,  the  function  is  such  that  any  and  every  path,  leading 
from  one  point  a  to  another  point  z  by  the  construction  of  a  series  of 
successive  domains  of  points  along  the  path,  gives  a  single  value  at  z  as  the 
continuation  of  one  initial  value  at  a.  When,  therefore,  there  is  only  a 
single  value  of  the  function  at  a,  the  process  of  continuation  leads  to  only  a 
single  value  of  the  function  at  any  other  point  in  the  plane.  The  function  is 
uniform  throughout  its  region  of  continuity.  The  detailed  properties  of  such 
functions  have  been  considered  in  the  preceding  chapters. 

In  the  second  case,  the  function  is  such  that  different  paths,  leading  from 
a  to  z,  do  not  give  a  single  value  at  z  as  the  continuation  of  one  and  the 
same  initial  value  at  a.  There  are  different  sets  of  elements  of  the  function, 
associated  with  different  sets  of  consecutive  domains  of  points  on  paths  from 
a  to  z,  which  lead  to  different  values  of  the  function  at  z;  but  any  change 
in  a  path  from  a  to  z  does  not  necessarily  cause  a  change  in  the  value  of  the 
function  at  z.  The  function  is  multiform  in  its  region  of  continuity.  The 
detailed  properties  of  such  functions  will  now  be  considered. 

90.  In  order  that  the  process  of  continuation  may  be  completely  carried 
out,  continuations  must  be  effected,  beginning  at  the  domain  of  any  point  a 
and  proceeding  to  the  domain  of  any  other  point  b  by  all  possible  paths  in 
the  region  of  continuity,  and  they  must  be  effected  for  all  points  a  and  b. 
Continuations  must  be  effected,  beginning  in  the  domain  of  every  point  a 
and  returning  to  that  domain  by  all  possible  closed  paths  in  the  region  of 
continuity.  When  they  are  effected  from  the  domain  of  one  point  a  to  that 
of  another  point  b,  all  the  values  at  any  point  z  in  the  domain  of  a  (and  not 
merely  a  single  value  at  such  points)  must  be  continued :  and  similarly  when 
they  are  effected,  beginning  in  the  domain  of  a  and  returning  to  that  domain. 
The  complete  region  of  the  plane  will  then  be  obtained  in  which  the  function 
can  be  represented  by  a  series  of  positive  integral  powers :  and  the  boundary 
of  that  region  will  be  indicated. 


90.]  OF  A   MULTIFORM   FUNCTION  151 

In  the  first  instance,  let  the  boundary  of  the  region  be  constituted  by  a 
number,  either  finite  or  infinite,  of 
isolated  points,  say  L1}  L2,  Ls,  ... 
Take  any  point  A  in  the  region,  so 
that  its  distance  from  any  of  the 
points  L  is  not  infinitesimal ;  and 
in  the  region  draw  a  closed  path 
ABC...EFA  so  as  to  enclose  one 
point,  say  Ll}  but  only  one  point,  of 
the  boundary  and  to  have  no  point 

of  the  curve  at  a  merely  infinitesimal  distance  from  L^  Let  such  curves  be 
drawn,  beginning  and  ending  at  A,  so  that  each  of  them  encloses  one  and 
only  one  of  the  points  of  the  boundary :  and  let  Kr  be  the  curve  which 
encloses  the  point  Lr. 

Let  Wj  be  one  of  the  power-series  defining  the  function  in  a  domain  with 
its  centre  at  A  :  let  this  series  be  continued  along  each  of  the  curves  Ks  by 
successive  domains  of  points  along  the  curve  returning  to  A.  The  result 
of  the  description  of  all  the  curves  will  be  that  the  series  w^  cannot  be 
reproduced  at  A  for  all  the  curves  though  it  may  be  reproduced  for  some 
of  them ;  otherwise,  w:  would  be  a  uniform  function.  Suppose  that  w.2,  w3> ..., 
each  in  the  form  of  a  power-series,  are  the  aggregate  of  new  distinct  values 
thus  obtained  at  A  ;  let  the  same  process  be  effected  on  w2,  w3,  ...  as  has 
been  effected  on  w1;  and  let  it  further  be  effected  on  any  new  distinct  values 
obtained  at  A  through  w2,  w3,  ...  ,  and  so  on.  When  the  process  has 
been  carried  out  so  far  that  all  values  obtained  at  A,  by  continuing  any 
series  round  any  of  the  curves  K  back  to  A,  are  included  in  values  already 
obtained,  the  aggregate  of  the  values  of  the  function  at  A  is  complete :  they 
are  the  values  at  A  of  the  branches  of  the  function. 

We  shall  now  assume  that  the  number  of  values  thus  obtained  is  finite, 
say  n,  so  that  the  function  has  n  branches  at  A  :  if  their  values  be  denoted 
by  w1}  w2,  ...,  wn,  these  n  quantities  are  all  the  values  of  the  function  at  A. 
Moreover,  n  is  the  same  for  all  points  in  the  plane,  as  may  be  seen  by  con 
tinuing  the  series  at  A  to  any  other  point  and  taking  account  of  the  corollaries 
at  the  end  of  the  present  section. 

The  boundary-points  L  may  be  of  two  kinds.  It  may  (and  not  infre 
quently  does)  happen  that  a  point  Ls  is  such  that,  whatever  branch  is  taken 
at  A  as  the  initial  value  for  the  description  of  the  circuit  Ks,  that  branch  is 
reproduced  at  the  end  of  the  circuit.  Let  the  aggregate  of  such  points  be 
/u  J2,  ....  Then  each  of  the  remaining  points  L  is  such  that  a  description 
of  the  circuit  round  it  effects  a  change  on  at  least  one  of  the  branches,  taken 
as  an  initial  value  for  the  description  ;  let  the  aggregate  of  these  points  be 
Blt  52,  ....  They  are  the  branch-points;  their  association  with  the  definition 
in  §  12  will  be  made  later. 


152 


DEFORMATION   OF   PATH 


[90. 


Fig.  15. 


When  account  is  taken  of  the  continuations  of  the  function  from  a  point 
A  to  another  point  B,  we  have  n  values  at  B  as  the  continuations  of  n  values 
at  A.  The  selection  of  the  individual  branch  at  B,  which  is  the  continuation 
of  a  particular  branch  at  A,  depends  upon  the  path  of  z  between  A  and  5; 
it  is  governed  by  the  following  fundamental  proposition : — 

The  final  value  of  a  branch  of  a  function  for  two  paths  of  variation  of  the 
independent  variable  from  one  point  to  another  will  be  the  same,  if  one  path 
can  be  deformed  into  the  other  without  passing  over  a  branch-point. 

Let  the  initial  and  the  final  points  be  a  and  b,  and  let  one  path  of 
variation  be  acb.  Let  another  path  of  variation  be  aeb, 
both  paths  lying  in  the  region  in  which  the  function  can 
be  expressed  by  series  of  positive  integral  powers :  the  two 
paths  are  assumed  to  have  no  point  within  an  infinitesimal 
distance  of  any  of  the  boundary-points  L  and  to  be  taken 
so  close  together  that  the  circles  of  convergence  of  pairs  of 
points  (such  as  cx  and  e1}  c2  and  e2,  and  so  on)  along  the  two 
paths  have  common  areas.  When  we  begin  at  a  with  a 
branch  of  the  function,  values  at  d  and  at  e^  are  obtained, 
depending  upon  the  values  of  the  branch  and  its  derivatives  at  a  and  upon 
the  positions  of  ca  and  e^  hence,  at  any  point  in  the  area  common  to  the 
circles  of  convergence  of  these  two  points,  only  a  single  value  arises  as 
derived  through  the  initial  value  at  a.  Proceeding  in  this  way,  only  a  single 
value  is  obtained  at  any  point  in  an  area  common  to  the  circles  of  con 
vergence  of  points  in  the  two  paths.  Hence  ultimately  one  and  the  same 
value  will  be  obtained  at  b  as  the  continuation  of  the  value  of  the  one  branch 
at  a  by  the  two  different  paths  of  variation  which  have  been  taken  so  that 
no  boundary-point  L  lies  between  them  or  infinitesimally  near  to  them. 

Now  consider  any  two  paths  from  a  to  b,  say  acb  and  adb,  such  that 
neither  of  them  is  near  a  boundary-point  and  that  the 
contour  they  constitute  does  not  enclose  a  boundary-point. 
Then  by  a  series  of  successive  infinitesimal  deformations  we 
can  change  the  path  acb  to  adb ;  and  as  at  b  the  same  value 
of  w  is  obtained  for  variations  of  z  from  a  to  b  along  the 
successive  deformations,  it  follows  that  the  same  value  of  w 
is  obtained  at  b  for  variations  of  z  along  acb  as  for  varia 
tions  along  adb. 

Next,  let  there  be  two  paths  acb,  adb  constituting  a  closed  contour, 
enclosing  one  (but  not  more  than  one)  of  the  points  /  and  none  of  the  points 
B.  When  the  original  curve  K  which  contains  the  point  /  is  described,  the 
initial  value  is  restored :  and  hence  the  branches  of  the  function  obtained  at 
any  point  of  K  by  the  two  paths  from  any  point,  taken  as  initial  point,  are 
the  same.  By  what  precedes,  the  parts  of  this  curve  K  can  be  deformed 


Fig.  16. 


90.]  OF   THE    VARIABLE  153 

into  the  parts  of  acbda  without  affecting  the  branches  of  the  function :  hence 
the  value  obtained  at  b,  by  continuation  along  acb,  is  the  same  as  the  value 
there  obtained  by  continuation  along  adb.  It  therefore  follows  that  a  path 
between  two  points  a  and  b  can  be  deformed  over  any  point  /  without 
affecting  the  value  of  the  function  at  b ;  so  that,  when  the  preceding 
results  are  combined,  the  proposition  enunciated  is  proved. 

By  the  continued  application  of  the  theorem,  we  are  led  to  the  following 
results : — 

COROLLARY  I.  Whatever  be  the  effect  of  the  description  of  a  circuit  on  the 
initial  value  of  a  function,  a  reversal  of  the  circuit  restores  the  original  value 
of  the  function. 

For  the  circuit,  when  described  positively  and  negatively,  may  be  re 
garded  as  the  contour  of  an  area  of  infinitesimal  breadth,  which  encloses  no 
branch-point  within  itself  and  the  description  of  the  contour  of  which 
therefore  restores  the  initial  value  of  the  function. 

COROLLARY  II.  A  circuit  can  be  deformed  into  any  other  circuit  without 
affecting  the  final  value  of  the  function,  provided  that  no  branch-point  be  crossed 
in  tJie  process  of  deformation. 

It  is  thus  justifiable,  and  it  is  often  convenient,  to  deform  a  path  con 
taining  a  single  branch-point  into  a  loop  round  the 
point.     A  loop*  consists  of  a  line  nearly  to  the  point,        °~ 
nearly  the  whole  of  a  very  small  circle  round  the  point,  Fig.  17. 

and  a  line  back  to  the  initial  point;  see  figure  17. 

COROLLARY  III.  The  value  of  a  function  is  unchanged  when  the  variable 
describes  a  closed  circuit  containing  no  branch-point ;  it  is  likewise  unchanged 
when  the  variable  describes  a  closed  circuit  containing  all  the  branch-points. 

The  first  part  is  at  once  proved  by  remarking  that,  without  altering  the 
value  of  the  function,  the  circuit  can  be  deformed  into  a  point. 

For  the  second  part,  the  simplest  plan  is  to  represent  the  variable  on 
Neumann's  sphere.  The  circuit  is  then  a  curve  on  the  sphere  enclosing  all 
the  branch-points  :  the  effect  on  the  value  of  the  function  is  unaltered  by  any 
deformation  of  this  curve  which  does  make  it  cross  a  branch-point.  The 
curve  can,  without  crossing  a  branch-point,  be  deformed  into  a  point  in  that 
other  part  of  the  area  of  the  sphere  which  contains  none  of  the  branch 
points  ;  and  the  point,  which  is  the  limit  of  the  curve,  is  not  a  branch 
point.  At  such  a  point,  the  value  of  the  function  is  unaltered  ;  and  there 
fore  the  description  of  a  circuit,  which  encloses  all  the  branch-points, 
restores  the  initial  value  of  the  function. 

COROLLARY  IV.     If  the  values  of  w  at  b  for  variations  along  two  paths 

*  French  writers  use  the  word  lacet,  German  writers  the  word  Schleife. 


154 


EFFECT   OF   DEFORMATION 


[90. 


acb,  adb  be  not  the  same,  then  a  description  of  acbda  will  not  restore  the  initial 
value  of  w  at  a. 

In  particular,  let  the  path  be  the  loop  OeceO  (fig.  17),  and  let  it  change  w 
at  0  into  w'.  Since  the  values  of  w  at  0  are  different  and  because  there  is 
no  branch-point  in  Oe  (or  in  the  evanescent  circuit  OeO),  the  values  of  w  at 
e  cannot  be  the  same :  that  is,  the  value  with  which  the  infinitesimal  circle 
round  a  begins  to  be  described  is  changed  by  the  description  of  that  circle. 
Hence  the  part  of  the  loop  that  is  effective  for  the  change  in  the  value  of  w  is 
the  small  circle  round  the  point ;  and  it  is  because  the  description  of  a  small 
circle  changes  the  value  of  w  that  the  value  of  w  is  changed  at  0  after  the 
description  of  a  loop. 

If/0?)  be  the  value  of  w  which  is  changed  mtof^z)  by  the  description  of 
the  loop,  so  that/Oz)  and  f^(z)  are  the  values  at  0,  then  the  foregoing 
explanation  shews  that  /(e)  and  /  (e)  are  the  values  at  e,  the  branch  /(e) 
being  changed  by  the  description  of  the  circle  into  the  branch  /i(e). 

From  this  result  the  inference  can  be  derived  that  the  points  Bl}  B.2,  ... 
are  branch-points  as  defined  in  §  12.  Let  a  be  any  one  of  the  points,  and 
let  f(z)  be  the  value  of  w  which  is  changed  into  f,  (z)  by  the  description  of 
a  very  small  circle  round  a.  Then  as  the  branch  of  w  is  monogenic,  the 
difference  between  f(z)  and  f^(z)  is  an  infinitesimal  quantity  of  the  same 
order  as  the  length  of  the  circumference  of  the  circle  :  so  that,  as  the  circle 
is  infinitesimal  and  ultimately  evanescent,  \f(z)  -/iOz)|  can  be  made  as  small 
as  we  please  with  decrease  of  z  -  a  or,  in  the  limit,  the  values  of  /(a)  and 
/(a)  at  the  branch-point  are  equal.  Hence  each  of  the  points  B  is  such 
that  two  or  more  branches  of  the  function  have  the  same  value  at  the  point 
and  there  is  interchange  among  these  branches  when  the  variable  describes  a 
small  circuit  round  the  point :  which  affords  a  definition  of  a  branch-point, 
more  complete  than  that  given  in  §  12. 

COROLLARY  V.  If  a  closed  circuit  contain  several  branch-points,  the  effect 
which  it  produces  can  be  obtained  by  a  combination  of  the  effects  produced  in 
succession  by  a  set  of  loops  each  going  round  only  one  of  the  branch-points. 

If  the  circuit  contain  several  branch-points,  say  three  as  at  a,  b,  c,  then  a 
path  such  as  AEFD,  in  fig.  18,  can  without 
crossing  any  branch-point,  be  deformed  into  the 
loops  AaB,  BbC,  GcD;  and  therefore  the  complete 
circuit  AEFD  A  can  be  deformed  validly  into 
AaBbCcDA,  and  the  same  effect  will  be  produced 
by  the  two  forms  of  circuit.  When  D  is  made  DA 

practically  to  coincide  with  A,  the  whole  of  the  Fig.  18. 

second  circuit  is  composed  of  the  three  loops.     Hence  the  corollary. 

This  corollary  is  of  especial  importance  in  the  consideration  of  integrals 
of  multiform  functions. 


91.]  OF   PATH   OF   THE   VAKIABLE  155 

COROLLARY  VI.  In  a  continuous  part  of  the  plane  where  there  are  no 
branch-points,  each  branch  of  a  multiform  function  is  uniform. 

Each  branch  is  monogenic  and,  except  at  isolated  points,  continuous; 
hence,  in  such  regions  of  the  plane,  all  the  propositions  which  have  been 
proved  for  monogenic  analytic  functions  can  be  applied  to  each  of  the 
branches  of  a  multiform  function. 

91.  If  there  be  a  branch-point  within  the  circuit,  then  the  value  of  the 
function  at  6  consequent  on  variations  along  acb  may,  but  will  not  necessarily, 
differ  from  its  value  at  the  same  point  consequent  on  variations  along  adb. 
Should  the  values  be  different,  then  the  description  of  the  whole  curve  acbda 
will  lead  at  a  not  to  the  initial  value  of  w,  but  to  a  different  value. 
The  test  as  to  whether  such  a  change  is  effected  by  the  description  is 
immediately  derivable  from  the  foregoing  proposition;  and  as  in  Corollary 
IV.,  §  90,  it  is  proved  that  the  value  is  or  is  not  changed  by  the  loop, 
according  as  the  value  of  w  for  a  point  near  the  circle  of  the  loop  is 
or  is  not  changed  by  the  description  of  that  circle.  Hence  it  follows  that,  if 
there  be  a  branch-point  which  affects  the  branch  of  the  function,  a  path  of 
variation  of  the  independent  variable  cannot  be  deformed  across  the  branch 
point  without  a  change  in  the  value  of  w  at  the  extremity  of  the  path. 

And  it  is  evident  that  a  point  can  be  regarded  as  a  branch-point  for  a 
function  only  if  a  circuit  round  the  point  interchange  some  (or  all)  of  the 
branches  of  the  function  which  are  equal  at  the  point.  It  is  not  necessary  that 
all  the  branches  of  the  function  should  be  thus  affected  by  the  point :  it  is 
sufficient  that  some  should  be  interchanged*. 

Further,  the  change  in  the  value  of  w  for  a  single  description  of  a  circuit 
enclosing  a  branch-point  is  unique. 

For,  if  a  circuit  could  change  w  into  w'  or  w",  then,  beginning  with  w" 
and  describing  it  in  the  negative  sense  we  should  return  to  w  and  afterwards 
describing  it  in  the  positive  sense  with  w  as  the  initial  value  we  should 
obtain  w'.  Hence  the  circuit,  described  and  then  reversed,  does  not  restore 
the  original  value  w"  but  gives  a  different  branch  w' ;  and  no  point  on 
the  circuit  is  a  branch-point.  This  result  is  in  opposition  to  Corollary  I., 
of  §  90 ;  and  therefore  the  hypothesis  of  alternative  values  at  the  end  of 
the  circuit  is  not  valid,  that  is,  the  change  for  a  single  description  is 
unique. 

But  repetitions  of  the  circuit  may,  of  course,  give  different  values  at  the 
end  of  successive  descriptions. 

*  In  what  precedes,  certain  points  were  considered  which  were  regular  singularities  (see 
p.  163,  note)  and  certain  which  were  branch-points.  Frequently  points  will  occur  which  are  at 
once  branch-points  and  infinities  ;  proper  account  must  of  course  be  taken  of  them. 


156 


LAW   OF    INTERCHANGE 


[92. 


Fig.  19. 


92.  Let  0  be  any  ordinary  point  of  the  function ;  join  it  to  all  the 
branch-points  (generally  assumed  finite  in 
number)  in  succession  by  lines  which  do  not 
meet  each  other :  then  each  branch  is  uniform 
for  each  path  of  variation  of  the  variable  which 
meets  none  of  these  lines.  The  effects  pro 
duced  by  the  various  branch-points  and  their 
relations  on  the  various  branches  can  be  indi 
cated  by  describing  curves,  each  of  which 
begins  at  a  point  indefinitely  near  0  and 
returns  to  another  point  indefinitely  near  it 
after  passing  round  one  of  the  branch- points, 
and  by  noting  the  value  of  each  branch  of  the  function  after  each  of  these 
curves  has  been  described. 

The  law  of  interchange  of  branches  of  a  function  after  description  of  a 
circuit  round  a  branch-point  is  as  follows: — 

All  the  branches  of  a  function,  which  are  affected  by  a  branch-point  as  such, 
can  either  be  arranged  so  that  the  order  of  interchange  (for  description  of  a 
path  round  the  point)  is  cyclical,  or  be  divided  into  sets  in  each  of  which  the 
order  of  interchange  is  cyclical. 

Let  wlt  w.2>  w3)...  be  the  branches  of  a  function  for  values  of  z  near  a 
branch-point  a  which  are  affected  by  the  description  of  a  small  closed  curve 
C  round  a :  they  are  not  necessarily  all  the  branches  of  the  function,  but  only 
those  affected  by  the  branch-point. 

The  branch  w^  is  changed  after  a  description  of  C ;  let  w2  be  the  branch 
into  which  it  is  changed.  Then  w2  cannot  be  unchanged  by  C;  for  a  reversed 
description  of  C,  which  ought  to  restore  w1}  would  otherwise  leave  w.2  un 
changed.  Hence  w2  is  changed  after  a  description  of  (7;  it  may  be  changed 
either  into  w1  or  into  a  new  branch,  say  w3.  If  into  wlt  then  w-^  and  w2  form 
a  cyclical  set. 

If  the  change  be  into  w3,  then  w3  cannot  remain  unchanged  after  a 
description  of  C,  for  reasons  similar  to  those  that  before  applied  to  the 
change  of  w.2:  and  it  cannot  be  changed  into  w2,  for  then  a  reversed  de 
scription  of  G  would  change  wz  into  w.A,  and  it  ought  to  change  w2  into  w^ 
Hence,  after  a  description  of  C,  w3  is  changed  either  into  w^  or  into  a  new 
branch,  say  w4.  If  into  w1}  then  w1}  w2,  w3  form  a  cyclical  set. 

If  the  change  be  into  w4,  then  w4  cannot  remain  unchanged  after  a 
description  of  G ;  and  it  cannot  be  changed  into  w.2  or  ws,  for  by  a  reversal 
of  the  circuit  that  earlier  branch  would  be  changed  into  w4  whereas  it  ought 
to  be  changed  into  the  branch,  which  gave  rise  to  it  by  the  forward  descrip 
tion — a  branch  which  is  not  w4.  Hence,  after  a  description  of  C,  w4  is 
changed  either  into  w^  or  into  a  new  branch.  If  into  wlf  then  wj}  w.2,  w3,  w4 
form  a  cyclical  set. 


92.]  OF  BRANCHES  OF  A  FUNCTION  157 

If  w4  be  changed  into  a  new  branch,  we  proceed  as  before  with  that  new 
branch  and  either  complete  a  cyclical  set  or  add  one  more  to  the  set.  By 
repetition  of  the  process,  we  complete  a  cyclical  set  sooner  or  later. 

If  all  the  branches  be  included,  then  evidently  their  complete  system 
taken  in  the  order  in  which  they  come  in  the  foregoing  investigation  is  a 
system  in  which  the  interchange  is  cyclical. 

If  only  some  of  the  branches  be  included,  the  remark  applies  to  the  set 
constituted  by  them.  We  then  begin  with  one  of  the  branches  not  included 
in  that  set  and  evidently  not  inclusible  in  it,  and  proceed  as  at  first,  until 
we  complete  another  set  which  may  include  all  the  remaining  branches  or 
only  some  of  them.  In  the  latter  case,  we  begin  again  with  a  new  branch 
and  repeat  the  process ;  and  so  on,  until  ultimately  all  the  branches  are 
included.  The  whole  system  is  then  arranged  in  sets,  in  each  of  which  the 
order  of  interchange  is  cyclical. 

93.  The  analytical  test  of  a  branch-point  is  easily  obtained  by  con 
structing  the  general  expression  for  the  branches  of  a  function  which  are 
interchanged  there. 

Let  z  =  a  be  a  branch-point  where  n  branches  w1}  ^v2,...,  wn  are  cyclically 
interchanged.  Since  by  a  first  description  of  a  small  curve  round  a,  the 
branch  w1  changes  into  w2,  the  branch  w»  into  ws,  and  so  on,  it  follows  that 
by  r  descriptions  w1  is  changed  into  wr+l  and  by  n  descriptions  wl  reverts  to 
its  initial  value.  Similarly  for  each  of  the  branches.  Hence  each  branch 
returns  to  its  initial  value  after  n  descriptions  of  a  circuit  round  a  branch 
point  where  n  branches  of  the  function  are  interchangeable. 

Now  let  z  -  a  =  Zn ; 

then,  when  z  describes  circles  round  a,  Z  moves  in  a  circular  arc  round  its 
origin.  For  each  circumference  described  by  z,  the  variable  Z  describes 

-th  part  of  its  circumference;  and  the  complete  circle  is  described  by  Z 
round  its  origin  when  n  complete  circles  are  described  by  z  round  a.  Now 
the  substitution  changes  wr  as  a  function  of  z  into  a  function  of  Z,  say  into 
Wr;  and,  after  n  complete  descriptions  of  the  ^-circle  round  a,  wr  returns 
to  its  initial  value.  Hence,  after  the  description  of  a  ^-circle  round  its 
origin,  Wr  returns  to  its  initial  value,  that  is,  Z  =  0  ceases  to  be  a  branch 
point  for  Wr.  Similarly  for  all  the  branches  W. 

But  no  other  condition  has  been  associated  with  a  as  a  point  for  the 
function  w ;  and  therefore  Z  =  0  may  be  any  point  for  the  function  W,  that 
is,  it  may  be  an  ordinary  point,  or  a  singularity.  In  every  case  we  have  W 
a  uniform  function  of  Z  in  the  immediate  vicinity  of  the  origin  ;  and  therefore 
in  that  vicinity  it  can  be  expressed  in  the  form 


158  ANALYTICAL   EXPRESSION  [93. 

with  the  significations  of  P  and  G  already  adopted.  When  Z  is  an  ordinary 
point,  G  is  a  constant  or  zero ;  when  Z  is  an  accidental  singularity,  O  is  an 
algebraical  function ;  and,  when  Z  is  an  essential  singularity,  G  is  a  transcen 
dental  function. 

The  simpler  cases  are,  of  course,  those  in  which  the  form  of  G  is  alge 
braical  or  constant  or  zero ;  and  then  W  can  be  put  into  the  form 

ZmP(Z), 

where  P  is  an  infinite  series  of  positive  powers  and  m  is  an  integer.  As  this 
is  the  form  of  W  in  the  vicinity  of  Z=Q,  it  follows  that  the  form  of  w  in  the 
vicinity  of  z  =  a  is 

m  1 

(z  -  a)n  P  {(z  -  a)n} 

and  the  various  n  branches  of  the  function  are  easily  seen  to  be  given  by 

i 
substituting  in  the  above  for  (z  —  a)n  the  values 

2im  j. 

e  m  (z  —  of, 

where  s  =  0,  1,...,  n  —  1.  We  therefore  infer  that  the  general  expression  for 
the  n  branches  of  a  function,  which  are  interchanged  by  circuits  round  a 
branch-point  z  =  a,  assumed  not  to  be  an  essential  singularity,  is 

m  _  1 

(z  -  a)Tl  P  {(z  -  a)»}, 

i 

where  m  is  an  integer,  and  where  to  (z  —  a)n  its  n  values  are  in  turn  assigned 
to  obtain  the  different  branches  of  the  function. 

There  may  be,  however,  more  than  one  cyclical  set  of  branches.     If  there 
be  another  set  of  r  branches,  then  it  may  similarly  be  proved  that  their 

general  expression  is 

OTI  _  i 

(zjaYQ{(z-ay-}, 

where  m^  is  an  integer,  and  Q  is  an  integral  function ;  the  various  branches 

i 
are  obtained  by  assigning  to  (z  —  a)r  its  r  values  in  turn. 

And  so  on,  for  each  of  the  sets,  the  members  of  which  are  cyclically 
interchangeable  at  the  branch-point. 

When  the  branch-point  is  at  infinity,  a  different  form  is  obtained.     Thus 
in  the  case  of  a  set  of  n  cyclically  interchangeable  branches  we  take 

z  =  %-», 

so  that  n  negative  descriptions  of  a  closed  £-curve,  excluding  infinity  and  no 
other  branch-point,  requires  a  single  positive  description  of  a  closed  curve 
round  the  w-origin.  These  n  descriptions  restore  the  value  of  w;  as  a  function 
of  z  to  its  initial  value;  and  therefore  the  single  description  of  the  M- curve 
round  the  origin  restores  the  value  of  U — the  equivalent  of  w  after  the 


93.]  NEAR   A   BRANCH-POINT  159 

change  of  the  independent  variable  —  as  a  function  of  u.     Thus  u  =  0  ceases 
to  be  a  branch-point  for  the  function  U  ;  and  therefore  the  form  of  U  is 


.  . 

where  the  symbols  have  the  same  general  signification  as  before. 

If,  in  particular,  z  =  oo  be  a  branch-point  but  not  an  essential  singularity, 
then  G  is  either  a  constant  or  an  algebraical  function  ;  and  then  U  can  be 
expressed  in  the  form 

u~mP(u}, 

where  TO  is  an  integer.     When  the  variable  is  changed  from  u  to  z,  then  the 
general  expression  for  the  n  branches  of  a  function  which  are  interchangeable 

at  z  =  oo  ,  assumed  not  to  be  an  essential  singularity,  is 


where  TO  is  an  integer  and  where  to  zn  its  n  values  are  assigned  to  obtain  the 
different  branches  of  the  function. 

If,  however,  the  branch-point  z  =  a  in  the  former  case  or  z  =  oo  in  the 
latter  be  an  essential  singularity,  the  forms  of  the  expressions  in  the  vicinity 
of  the  point  are 

_i  i 

G{(z-a)  »J>-p{(jr-aJ»}f 

i  _i 

and  G(zn)  +  P(z  n), 

respectively. 

Note.  When  a  multiform  function  is  denned,  either  explicitly  or  im 
plicitly,  it  is  practically  always  necessary  to  consider  the  relations  of  the 
branches  of  the  function  for  z  =  oo  as  well  as  their  relations  for  points  that 
are  infinities  of  the  function.  The  former  can  be  determined  by  either 
of  the  processes  suggested  in  §  4  for  dealing  with  z=<x>;  the  latter  can  be 
determined  as  in  the  present  article. 

Moreover,  the  total  number  of  branches  of  the  function  has  been  assumed 
to  be  finite.  The  cases,  in  which  the  number  of  branches  is  unlimited,  need 
not  be  discussed  in  general  :  it  will  be  sufficient  to  consider  them  when  they 
arise,  as  they  do  arise,  e.g.,  when  the  function  is  of  the  form  of  an  algebraical 
irrational  with  an  irrational  index  such  as  z^  —  hardly  a  function  in  the 
ordinary  sense—,  or  when  the  function  is  the  logarithm  of  a  function  of  z, 
or  is  the  inverse  of  a  periodic  function.  In  the  nature  of  their  multiplicity 
of  branching  and  of  their  sequence  of  interchange,  they  are  for  the  most  part 
distinct  from  the  multiform  functions  with  only  a  finite  number  of  branches. 

Ex.     The   simplest  illustrations  of  multiform  functions  are   furnished   by  functions 
denned  by  algebraical  equations,  in  particular,  by  algebraic  irrationals. 


160 


ALGEBRAICAL 


[93. 


The  general  type  of  the  algebraical  irrational  is  the  product  of  a  number  of  functions 
of  the  form  w  =  {A(z  —  al)(z-a.2)  ......  (z-a^)}m,  m  and  n  being  integers. 

This  particular  function  has  m  branches;  the  points  a1}  «2,  ......  ,  an  are  branch-points. 

To  find  the  law  of  interchange,  we  take  z-ar  =  pe01;  then  when  a  small  circle  of  radius  p 
is  described  round  ar,  so  that  z  returns  to  its  initial  position,  the  value  of  6  increases  by 

2n  and  the  new  value  of  w  is  aw,  where  a  is  the  with  root  of  unity  defined  by  em  m.    Taking 
then  the  various  branches  as  given  by  w,  aw,  a?w,  ......  ,  am~lw,  we  have  the  law  of  inter 

change  for  description  of  a  small  curve  round  any  one-branch  point  as  given  by  this 
succession  in  cyclical  order.  The  law  of  succession  for  a  circuit  enclosing  more  than 
one  of  the  branch-points  is  derivable  by  means  of  Corollary  V,  §  90. 

To  find  the  relation  of  z  =  o>  to  w,  we  take  zz'  =  l  and  consider  the  new  function  W  in 
the  vicinity  of  the  ^'-origin.  We  have 

W  ={A  (1  -VH1  -a/)  ......  (l-an<)}^'~»». 

If  the  variable  z1  describe  a  very  small  circle  round  the  origin  in  the  negative  sense,  then 

27TZ  — 

z'  is  multiplied  by  e~2™  and  so  W  acquires  a  factor  e     ™,  that  is,  W  is  changed  unless 
this  acquired  factor  is  unity.     It  can  be  unity  only  when  n/m  is  an  integer  ;  and  therefore, 
except   when  n/m  is  an  integer,  0=00    is  a  branch-point  of  the  function.     The  law  of 
succession   is   the   same  as  that  for   negative  description  of  the  z'-circle,   viz.,   w,   anw, 
a2nw,  ......  ;  the  m  values  form  a  single  cycle  only  if  n  be  prime  to  m,  and  a  set  of  cycles 

if  n  be  not  prime  to  m. 

Thus  0=00  is  a  branch  -point  for  w  =  (k?-gg-g^~^  ;  it  is  not  a  branch-point  for 
w  =  {(\  -22)  (1  —  &2z2)}~*;  and  z  =  b  is  a  branch-point  for  the  function  defined  by 

(z  —  b)  w2  =  z  —  a, 
but  z  =  b  is  riot  a  branch-point  for  the  function  defined  by  (z—b)2wz  =  z-a. 

Again,  if  p  denote  a  particular  value  of  ft  when  z  has  a  given  value,  and  q  similarly 

denote  a  particular  value  of  [—  —  :  )  ,  then  w=p+q  is  a  six-valued  function,  the  values 

V+v 
being 


W6=  -p  +  aq, 

where  a  is  a  primitive  cube  root  of  unity.  The  branch-points  are  -  1,  0,  1,  oo  ;  and  the 
orders  of  change  for  small  circuits  round  one  (and  only  one)  of  these  points  are  as 
follows  : 


For  a  small  circuit  round 

-1 

0 

1 

00 

Wj  changes  to 

•ft 

W-i 

W3 

W2 

W2               „ 

we 

W1 

W4 

w, 

^3 

to, 

W4 

W5 

W4 

W4              „ 

M>2 

w, 

W6 

W3 

Ws             n 

W3 

W6 

w, 

W6 

U>6 

W74 

W5 

Wo 

W5 

93.]  FUNCTIONS  161 

Combinations  can  at  once  be  effected  ;  thus,  for  a  positive  circuit  enclosing  both  1  and  QO 
but*  not  —  1  or  0,  the  succession  is 

iolt  w4,  w6,  w2,  w3,  WG 
in  cyclical  order. 

94.  It  has  already  been  remarked  that  algebraic  irrationals  are  a  special 
class  of  functions  denned  by  algebraical  equations.  Functions  thus  generally 
denned  by  equations,  which  are  algebraical  so  far  as  concerns  the  dependent 
variable  but  need  not  be  so  in  reference  to  the  independent  variable,  are 
often  called  algebraical.  The  term,  in  one  sense,  cannot  be  strictly  applied 
to  the  roots  of  an  equation  of  every  degree,  seeing  that  the  solution 
of  equations  of  the  fifth  and  higher  degrees  can  be  effected  only  by 
transcendental  functions;  but  what  is  implied  is  that  a  finite  number  of 
determinations  of  the  dependent  variable  is  given  by  the  equation  -f*. 

The  equation  is  algebraical  in  relation  to  the  dependent  variable  w,  that 
is,  it  will  be  taken  to  be  of  finite  degree  n  in  w.  The  coefficients  of  the 
different  powers  will  be  supposed  to  be  rational  uniform  functions  of  z  :  were 
they  irrational  in  any  given  equation,  the  equation  could  be  transformed 
into  another,  the  coefficients  of  which  are  rational  uniform  functions.  And 
the  equation  is  supposed  to  be  irreducible,  that  is,  if  the  equation  be  taken 
in  the  form 

f(w,  *)  =  0, 

the  left-hand  member  f(w,  z)  cannot  be  resolved  into  factors  of  a  form  and 
character  as  regards  w  and  z  similar  to  /itself. 

The  existence  of  equal  roots  of  the  equation  for  general  values  of  z 
requires  that 

fi       \       j    "df(w>  z) 
f(w,z)   and    ^~ 

shall  have  a  common  factor,  which  will  be  rational  owing  to  the  form  of 
f(w,  z}.  This  form  of  factor  is  excluded  by  the  irreducibility  of  the  equation  ; 
so  that  /=  0,  as  an  equation  in  w,  has  not  equal  roots  for  general  values 
of  z.  But  though  the  two  equations  are  not  both  satisfied  in  virtue  of  a 
simpler  equation,  they  are  two  equations  determining  values  of  w  and  #; 
and  their  form  is  such  that  they  will  give  equal  values  of  w  for  special 
values  of  z. 

Since  the  equation  is  of  degree  n,  it  may  be  taken  to  be 


w 


where  the  functions  F1}  F2}...  are  rational  and  uniform.     If  all  their  singu- 

*  Such  a  circuit,  if  drawn  on  the  Neumann's  sphere,  may  be  regarded  as  excluding  -  1  and  0, 
or  taking  account  of  the  other  portion  of  the  surface  of  the  sphere,  it  may  be  regarded  as  a 
negative  circuit  including  -  1  and  0,  the  cyclical  interchange  for  which  is  easily  proved  to  be 
iCj,  w4,  w5,  w.2,  M?3,  w6  as  in  the  text. 

t  Such  a  function  is  called  Men  defini  by  Liouville. 

F.  11 


162  ALGEBRAICAL  [94. 

larities  be  accidental,  they  are  raeromorphic  algebraical  functions  of  z  (unless 
z  =  oo  is  the  only  singularity,  in  which  case  they  are  holomorphic)  ;  and  the 
equation  can  then  be  replaced  by  one  which  is  equivalent  and  has  all  its 
coefficients  holomorphic,  the  coefficient  of  wn  being  the  least  common  multiple 
of  all  the  denominators  of  the  meromorphic  functions  in  the  first  form.  This 
form  cannot  however  be  deduced,  if  any  of  the  singularities  be  essential. 

The  equation,  as  an  equation  in  w,  has  n  roots,  all  functions  of  z  ;  let 
these  be  denoted  by  w1,w2,...,  ivn,  which  are  the  n  branches  of  the  function  w. 
When  the  geometrical  interpretation  is  associated  with  the  analytical  relation, 
there  are  n  points  in  the  w-plane,  say  a1,...,  an,  which  correspond  with  a  point 
in  the  ^-plane,  say  with  c^  ;  and  in  general  these  n  points  are  distinct.  As 
z  varies  so  as  to  move  in  its  own  plane  from  a,  then  each  of  the  w-points 
moves  in  their  common  plane  ;  and  thus  there  are  n  w-paths  corresponding 
to  a  given  z-path.  These  n  curves  may  or  may  not  meet  one  another. 

If  they  do  not,  there  are  n  distinct  w-paths,  leading  from  a1;...,  an  to 
/3i,...,  /3n,  respectively  corresponding  to  the  single  ^-path  leading  from  a 
to  b. 

If  two  or  more  of  the  w-paths  do  meet  one  another,  and  if  the  describing 
w-poirits  coincide  at  their  point  of  intersection,  then  at  such  a  point  of 
intersection  in  the  w-plane,  the  associated  branches  w  are  equal  ;  and 
therefore  the  point  in  the  ^-plane  is  a  point  that  gives  equal  values  for  w. 
It  is  one  of  the  roots  of  the  equation  obtained  by  the  elimination  of  w 
between 


the  analytical  test  as  to  whether  the  point  is  a  branch-point  will  be 
considered  later.  The  march  of  the  concurrent  ^-branches  from  such  a 
point  of  intersection  of  two  w-paths  depends  upon  their  relations  in  its 
immediate  vicinity. 

When  no  such  point  lies  on  a  ^-path  from  a  to  b,  no  two  of  the  w-points 
coincide  during  the  description  of  their  paths.  By  §  90,  the  2-path  can  be 
deformed  (provided  that,  in  the  deformation,  it  does  not  cross  a  branch-point) 
without  causing  any  two  of  the  w-points  to  coincide.  Further,  if  z  describe 
a  closed  curve  which  includes  none  of  the  branch  -points,  then  each  of  the 
^-branches  describes  a  closed  curve  and  no  two  of  the  tracing  points  ever 
coincide. 

Note.  The  limitation  for  a  branch-point,  that  the  tracing  w-points 
coincide  at  the  point  of  intersection  of  the  w-curves,  is  of  essential  im 
portance. 

What  is  required  to  establish  a  point  in  the  z-plane  as  a  branch-point, 
is  not  a  mere  geometrical  intersection  of  a  couple  of  completed  w-paths  but 
the  coincidence  of  the  w-points  as  those  paths  are  traced,  together  with  inter- 


94.]  FUNCTIONS  1  63 

change  of  the  branches  for  a  small  circuit  round  the  point.  Thus  let  there  be 
such  a  geometrical  intersection  of  two  w-curves,  without  coincidence  of  the 
tracing  points.  There  are  two  points  in  the  ^-plane  corresponding  to  the 
geometrical  intersection  ;  one  belongs  to  the  intersection  as  a  point  of  the 
w-paih  which  first  passed  through  it,  and  the  other  to  the  intersection  as  a 
point  of  the  w-path  which  was  the  second  to  pass  through  it.  The  two 
branches  of  w  for  the  respective  values  of  z  are  undoubtedly  equal  ;  but  the 
equality  would  not  be  for  the  same  value  of  z.  And  unless  the  equality 
of  branches  subsists  for  the  same  value  of  z,  the  point  is  not  a  branch 
point. 

A  simple  example  will  serve  to  illustrate  these  remarks.     Let  w  be  defined  by  the 
equation 


so  that  the  branches  w1  and  w2  are  given  by 

Ci0j_  =  cz+z(z2  +  c2)*,         cw2  =  cz-z(z*-\-  c2)*  ; 
it  is  easy  to  prove  that  the  equation  resulting  from  the  elimination  of  w  between  /=0  and 


and  that  only  the  two  points  z=  ±ic  are  branch-points. 

The  values  of  z  which  make  wl  equal  to  the  value  of  wz  for  z  =  a  (supposed  not  equal  to 
either  0,  ci  or  —  ci)  are  given  by 

cz  +  z  (02  +  c2)*  =  ca  -  a  (a2  +  c2)*, 

which  evidently  has  not  2  =  a  for  a  root.  Rationalising  the  equation  so  far  as  concerns  z 
and  removing  the  factor  z  -a,  as  it  has  just  been  seen  not  to  furnish  a  root,  we  find  that  s 
is  determined  by 

z3  +  z2a  +  za2  +  a3  +  2ac2  -  2ac  (a2  +  c2)  *  =  0, 

the  three  roots  of  which  are  distinct  from  a,  the  assumed  point,  and  from  ±ci,  the  branch 
point.  Each  of  these  three  values  of  z  will  make  wv  equal  to  the  value  of  w2  for  z=a  :  we 
have  geometrical  intersection  without  coincidence  of  the  tracing  points. 

95.  When  the  characteristics  of  a  function  are  required,  the  most  im 
portant  class  are  its  infinities:  these  must  therefore  now  be  investigated. 
It  is  preferable  to  obtain  the  infinities  of  the  function  rather  than  the 
singularities  alone,  in  the  vicinity  of  which  each  branch  of  the  function 
is  uniform  *  :  for  the  former  will  include  these  singularities  as  well  as 
those  branch-points  which,  giving  infinite  values,  lead  to  regular  singularities 
when  the  variables  are  transformed  as  in  §  93.  The  theorem  which  deter 
mines  them  is:  — 

The  infinities  of  a  function  determined  by  an  algebraical  equation  are  the 
singularities  of  the  coefficients  of  the  equation. 
Let  the  equation  be 

wn  +  wn-i  FI  ^  +  wn-,  !»,(*)  +  ...  +  rf^  (^)  +  ^  (^)  =  Q, 

*  These  singularities  will,  for  the  sake  of  brevity,  be  called  regular. 

11—2 


164  INFINITIES  [95. 

and  let  w'  be  any  branch  of  the  function;   then,  if  the  equation  which 
determines  the  remaining  branches  be 

wn-i  +  wn-2  Qi  ^  +  wn-3  £2  (Y)  +  . . .  +  WGn-z  (Z)  +  Gn-i  (z)  =  0, 

we  have  Fn  (z)  =  -  w'Gn-i  (z), 

Fn^  (z)  =  -  w'Gn-z  (z)  +  £„_!  (z), 

^71-2  (Z)  =  -  w'Gn-s  (z}  +  #n-2  (z), 


Now  suppose  that  a  is  an  infinity  of  w' ;  then,  unless  it  be  a  zero  of  order 
at  least  equal  to  that  of  Gn^  (z),  a  is  an  infinity  of  Fn  (z).  If,  however,  it  be 
a  zero  of  Gn-i  (z)  of  sufficient  order,  then  from  the  second  equation  it  is  an 
infinity  of  Fn_l(z)  unless  it  is  a  zero  of  order  at  least  equal  to  that  of 
6rn_2  (z) ;  and  so  on.  The  infinity  must  be  an  infinity  of  some  coefficient  not 
earlier  than  Fi  (z)  in  the  equation,  or  it  must  be  a  zero  of  all  the  functions 
G  which  are  later  than  Gf_!  (z).  If  it  be  a  zero  of  all  the  functions  Gr,  so 
that  we  may  not,  without  knowing  the  order,  assert  that  it  is  of  rank  at 
least  equal  to  its  order  as  an  infinity  of  w',  still  from  the  last  equation  it 
follows  that  a  must  be  an  infinity  of  Fl  (z).  Hence  any  infinity  of  w  is  an 
infinity  of  at  least  one  of  the  coefficients  of  the  equation. 

Conversely,  from  the  same  equations  it  follows  that  a  singularity  of  one 
of  the  coefficients  is  an  infinity  either  of  w'  or  of  at  least  one  of  the  co 
efficients  G.  Similarly  the  last  alternative  leads  to  an  inference  that  the 
infinity  is  either  an  infinity  of  another  branch  w"  or  of  the  coefficients  of  the 
(theoretical)  equation  which  survives  when  the  two  branches  have  been 
removed.  Proceeding  in  this  way,  we  ultimately  find  that  the  infinity  either 
is  an  infinity  of  one  of  the  branches  or  is  an  infinity  of  the  coefficient  in  the 
last  equation,  that  is,  of  the  last  of  the  branches.  Hence  any  singularity 
of  a  coefficient  is  an  infinity  of  at  least  one  of  the  branches  of  the  function. 

It  thus  appears  that  all  the  infinities  of  the  function  are  included  among, 
and  include,  all  the  singularities  of  the  coefficients ;  but  the  order  of  the 
infinity  for  a  branch  does  not  necessarily  make  that  point  a  regular 
singularity  nor,  if  it  be  made  a  regular  singularity,  is  the  order  necessarily 
the  same  as  for  the  coefficient. 

96.  The  following  method  is  effective  for  the  determination  of  the  order 
of  the  infinity  of  the  branch. 

Let  a  be  an  accidental  singularity  of  one  or  more  of  the  F  functions, 
say  of  order  ra;  for  the  function  Ft ;  and  assume  that,  in  the  vicinity  of  a, 
we  have 

Ft  (z)  =  (z-  a)-™*  [Ci  +  di  (z-a)  +  e{  (z  -  a?  +...]. 


96.] 


OF   ALGEBRAICAL   FUNCTIONS 


165 


Then  the  equation  which  determines  the  first  term  of  the  expansion  of  w  in 
a  series  in  the  vicinity  of  a  is 

wn  +  d  (z  —  a)~™i  wn~l  +  c2(z  —  a)~m2  wn~2  +  ... 

-f  cn_!  (z  -  a)~m»-i  w  +  cn  (z  -  a)~m«  =  0. 

Mark    in    a    plane,   referred  to   two  rectangular  axes,  points   n,  0;    n  —  1, 
—  m^;  n  —  2,  —  m2 ; . . .,  0,  —  mn ;  let  these 
be  A0,  A1} ...,  An  respectively.     Any  line 
through  Ai  has  its  equation  of  the  form 

1 1  —I—  nm  •  ~~  ~\  J  o"  —  l  w  — T.  t)\\ 
y  T  »*<  —  A,  {J,         (71,         tftf 

that  is, 

y  —  \x  =  —  \  (n  —  i)  —  mi. 

If  then  w  =  (z  —  a)~xf(z},  where  f(z)  is 
finite  when  z  =  a,  the  intercept  of  the  fore 
going  line  on  the  negative  side  of  the  axis  of  y  is  equal  to  the  order  of  the 
infinity  in  the  term 

wn-iFi(z). 

This  being  so,  we  take  a  line  through  An  coinciding  in  direction  with  the 
negative  part  of  the  axis  of  y  and  we  turn  it  about  An  in  a  trigonometrically 
positive  direction  until  it  first  meets  one  of  the  other  points,  say  An_r ;  then 
we  turn  it  about  An_r  until  it  meets  one  of  the  other  points,  say  An_s;  and 
so  on  until  it  passes  through  A0.  There  will  thus  be  a  line  from  An  to 
A0,  generally  consisting  of  a  number  of  parts ;  and  none  of  the  points  A 
will  be  outside  it. 

The  perpendicular  from  the  origin  on  the  line  through  An_r  and  An_g  is 
evidently  greater  than  the  perpendicular  on  any  parallel  line  through  a 
point  A,  that  is,  on  any  line  through  a  point  A  with  the  same  value 
of  X;  and,  as  this  perpendicular  is 

it  follows  the  order  of  the  infinite  terms  in  the  equation,  when  the  particular 
substitution  is  made  for  w,  is  greater  for  terms  corresponding  to  points  lying 
on  the  line  than  it  is  for  any  other  terms. 

If  /(*)  =  0  wnen  z  =  a,  then  the  terms  of  lowest  order  after  the  substitu 
tion  of  (z  —  a)~Kf(z)  for  w  are 

as  many  terms  occurring  in  the  bracket  as  there  are  points  A  on  the  line 
joining  An_r  to  An_s.  Since  the  equation  determining  w  must  be  satisfied, 
terms  of  all  orders  must  disappear,  and  therefore 


an  equation  determining  s-r  values  of  6,  that  is,  the  first  terms  in  the 
expansions  of  s  —  r  branches  w. 


166  INFINITIES  [96. 

Similarly  for  each  part  of  the  line  :  for  the  first  part,  there  are  r  branches 
with  an  associated  value  of  X  ;  for  the  second,  s  —  r  branches  with  another 
associated  value  ;  for  the  third,  t  —  s  branches  with  a  third  associated  value  ; 
and  so  on. 

The  order  of  the  infinity  for  the  branches  is  measured  by  the  tangent 
of  the  angle  which  the  corresponding  part  of  the  broken  line  makes  with  the 
axis  of  a;  ;  thus  for  the  line  joining  An^.  to  An_s  the  order  of  the  infinity  for 
the  s  -  r  branches  is 


where  mn_r  and  mn_s  are  the  orders  of  the  accidental  singularities  of  Fn_r  (z) 
and  Fn_s  (z). 

If  any  part  of  the  broken  line  should  have  its  inclination  to  the  axis  of 
x  greater  than  \ir  so  that  the  tangent  is  negative  and  equal  to  -  //,,  then  the 
form  of  the  corresponding  set  of  branches  w  is  (z  —  a,y  g  {z}  for  all  of  them, 
that  is,  the  point  is  not  an  infinity  for  those  branches.  But  when  the 
inclination  of  a  part  of  the  line  to  the  axis  is  <  \TT,  so  that  the  tangent  is 
positive  and  equal  to  X,  then  the  form  of  the  corresponding  set  of  branches 
w  is  (z  —  a)~Kf(z)  for  all  of  them,  that  is,  the  point  is  an  infinity  of  order  X 
for  those  branches. 

In  passing  from  An  to  A0  there  may  be  parts  of  the  broken  line  which 
have  the  tangential  coordinate  negative,  implying  therefore  that  a  is  not  an 
infinity  of  the  corresponding  set  or  sets  of  branches  w.  But  as  the  revolving 
line  has  to  change  its  direction  from  Any'  to  some  direction  through  A0, 
there  must  evidently  be  some  part  or  parts  of  the  broken  line  which  have 
their  tangential  coordinate  positive,  implying  therefore  that  a  is  an  infinity 
of  the  corresponding  set  or  sets  of  branches. 

Moreover,  the  point  a  is,  by  hypothesis,  an  accidental  singularity  of  at 
least  one  of  the  coefficients  and  it  has  been  supposed  to  be  an  essential 
singularity  of  none  of  them;  hence  the  points  A0,  A1}  ...,  An  are  all  in  the 
finite  part  of  the  plane.  And  as  no  two  of  their  abscissa  are  equal,  no  line 
joining  two  of  them  can  be  parallel  to  the  axis  of  y,  that  is,  the  inclination 
of  the  broken  line  is  never  \ir  and  therefore  the  tangential  coordinate  is 
finite,  that  is,  the  order  of  the  infinity  for  the  branches  is  finite  for  any 
accidental  singularity  of  the  coefficients. 

If  the  singularity  at  a  be  essential  for  some  of  the  coefficients,  the 
corresponding  result  can  be  inferred  by  passing  to  the  limit  which  is 
obtained  by  making  the  corresponding  value  or  values  of  m  infinite.  In 
that  case  the  corresponding  points  A  move  to  infinity  and  then  parts  of  the 
broken  line  pass  through  A0  (which  is  always  on  the  axis  of  x)  parallel  to 
the  axis  of  y,  that  is,  the  tangential  coordinate  is  infinite  and  the  order  of 


96.]  OF   ALGEBRAICAL   FUNCTIONS  167 

the  infinity  at  a  for  the  corresponding  branches  is  also  infinite.  The  point  is 
then  an  essential  singularity  (and  it  may  be  also  a  branch-point). 

It  has  been  assumed  implicitly  that  the  singularity  is  at  a  finite  point  in 
the  2-plane ;  if,  however,  it  be  at  oo ,  we  can,  by  using  the  transformation 
zz'  —  1  and  discussing  as  above  the  function  in  vicinity  of  the  origin,  obtain 
the  relation  of  the  singularity  to  the  various  branches.  We  thus  have  the 
further  proposition : 

The  order  of  ike  infinity  of  a  branch  of  an  algebraical  function  at  a 
singularity  of  a  coefficient  of  the  equation,  which  determines  the  function,  is 
finite  or  infinite  according  as  the  singularity  is  accidental  or  essential. 

If  the  coefficients  FI  of  the  equation  be  holomorphic  functions,  then 
z  =  oo  is  their  only  singularity  and  it  is  consequently  the  only  infinity  for 
branches  of  the  function.  If  some  of  or  all  the  coefficients  Ff  be  mero- 
morphic  functions,  the  singularities  of  the  coefficients  are  the  zeros  of 
the  denominators  and,  possibly,  £=oo;  and,  if  the  functions  be  algebraical, 
all  such  singularities  are  accidental.  In  that  case,  the  equation  can  be 
modified  to 

h0  (z)  wn  +  h^  (z}  wn~l  +  A2  (z)  wn~2  +  . . .  =  0, 

where  h0(z)  is  the  least  common  multiple  of  all  the  denominators  of  the 
functions  Ft.  The  preceding  results  therefore  lead  to  the  more  limited 
theorem  : 

When  a  function  w  is  determined  by  an  algebraical  equation  the  coefficients 
of  which  are  holomorphic  functions  of  z,  then  each  of  the  zeros  of  the  coefficient 
of  the  highest  power  of  w  is  an  infinity  of  some  of  (and  it  may  be  of  all)  the 
branches  of  the  function  w,  each  such  infinity  being  of  finite  order.  The  point 
z=  oo  may  also  be  an  infinity  of  the  function  w ;  the  order  of  that  infinity  is 
finite  or  infinite  according  as  z  =  oo  is  an  accidental  or  an  essential  singularity 
of  any  of  the  coefficients. 

It  will  be  noticed  that  no  precise  determination  of  the  forms  of  the 
branches  w  at  an  infinity  has  been  made.  The  determination  has,  however, 
only  been  deferred :  the  infinities  of  the  branches  for  a  singularity  of  the 
coefficients  are  usually  associated  with  a  branch-point  of  the  function  and 
therefore  the  relations  of  the  branches  at  such  a  point  will  be  of  a  general 
character  independent  of  the  fact  that  the  point  is  an  infinity. 

If,  however,  in  any  case  a  singularity  of  a  coefficient  should  prove  to  be, 
not  a  branch-point  of  w  but  only  a  regular  singularity,  then  in  the  vicinity  of 
that  point  the  branch  of  w  is  a  uniform  function.  A  necessary  (but  not  suffi 
cient)  condition  for  uniformity  is  that  (mn_r  —  mn_s)  -7-  (s  —  r)  be  an  integer. 

Note.  The  preceding  method  can  be  applied  to  determine  the  leading 
terms  of  the  branches  in  the  vicinity  of  a  point  a  which  is  an  ordinary  point 
for  each  of  the  coefficients  F. 


168  BRANCH-POINTS  [97. 

97.  There  remains  therefore  the  consideration  of  the  branch-points  of  a 
function  determined  by  an  algebraical  equation. 

The  characteristic  property  of  a  branch-point  is  the  equality  of  branches 
of  the  function  for  the  associated  value  of  the  variable,  coupled  with  the 
interchange  of  some  of  (or  all)  the  equal  branches  after  description  by  the 
variable  of  a  small  contour  enclosing  the  point. 

So  far  as  concerns  the  first  part,  the  general  indication  of  the  form  of  the 
values  has  already  (§  93)  been  given.  The  points,  for  which  values  of  w 
determined  as  a  function  of  z  by  the  equation 

f(w,  z)  =J0 

are  equal,  are  determined  by  the  solution  of  this  equation  treated  simul 
taneously  with 

df(w,  z)  =  Q. 
dw 

and  when  a  point  z  is  thus  determined  the  corresponding  values  of  w,  which 
are  equal  there,  are  obtained  by  substituting  that  value  of  z  and  taking  M, 

the  greatest  common  measure  of  /  and  -J-  .     The  factors  of  M  then  lead  to 

the  value  or  the  values  of  w  at  the  point  ;  the  index  m  of  a  linear  factor 
gives  at  the  point  the  multiplicity  of  the  value  which  it  determines,  and 
shews  that  m  +  1  values  of  w  have  a  common  value  there,  though  they  are 
distinct  at  infinitesimal  distances  from  the  point.  If  m  =  1  for  any  factor, 
the  corresponding  value  of  w  is  an  isolated  value  and  determines  a  branch 
that  is  uniform  at  the  point. 

Let  z  =  a,  w  =  a  be  a  value  of  z  and  a  value  of  w  thus  obtained  ;  and 
suppose  that  m  is  the  number  of  values  of  w  that  are  equal  to  one  another. 
The  point  z  =  a  is  not  a  branch-point  unless  some  interchange  among  the 
in  values  of  w  is  effected  by  a  small  circuit  round  a  ;  and  it  is  therefore 
necessary  to  investigate  the  values  of  the  branches*  in  the  vicinity  of  z  —  a. 

Let  w  =  a.  +  w',  z  =  a  +  z'  ;  then  we  have 


that  is,  on  the  supposition  that  f(w,  z)  has  been  freed  from  fractions, 

/(a,  a)  +  SS^rXV  =  0, 

r,  s 

so  that,  since  a  is  a  value  of  w  corresponding  to  the  value  a  of  z,  we  have 
w'  and  /  connected  by  the  relation 


*  The  following  investigations  are  founded  on  the  researches  of  Puiseux  on  algebraic 
functions;  they  are  contained  in  two  memoirs,  Liouville,  lre  Ser.,  t.  xv,  (1850),  pp.  365  —  480,  ib., 
t.  xvi,  (1851),  pp.  228—240.  See  also  the  chapters  on  algebraic  functions,  pp.  19  —  76,  in  the 
second  edition  of  Briot  and  Bouquet's  Theorie  des  fonctions  elliptiques. 


97.]  OF   ALGEBRAICAL   FUNCTIONS  169 

When  /  is  0,  the  zero  value  of  w'  must  occur  m  times,  since  a  is  a  root 
m  times  repeated;   hence  there  are  terms  in  the  foregoing  equation  inde 

pendent  of  z,  and  the  term  of  lowest  index  among  them  is  w'm.  Also  when 
w  '  =  0,  z'  —  0  is  a  possible  root  ;  hence  there  must  be  a  term  or  terms 
independent  of  w'  in  the  equation. 

First,  suppose  that  the  lowest  power  of  z  among  the  terms  independent 
of  w'  is  the  first.     The  equation  has  the  form 

Az'  +  higher  powers  of  z' 
+  Biu'    +  higher  powers  of  w' 
+  terms  involving  z'  and  w'  =  0, 

O-/*  /  \ 

where  A  is  the  value  of  -        '  —  -  for  w  =  a,  z  =  a.     Let  z'=%m,  w'  =  v%:  the 

02 

last  form  changes  to 

(A  +  Bvm)  £m  +  terms  with  £m+1  as  a  factor  =  0  ; 
and  therefore  A  +  Bvm  +  terms  involving  £=  0. 

Hence  in  the  immediate  vicinity  of  z  =  a,  that  is,  of  £  =0,  we  have 

A  +  Bv™  =  0. 

Neither  A  nor  B  is  zero,  so  that  all  the  m  values  of  v  are  finite.  Let  them 
be  vl}...,  vm,  so  arranged  that  their  arguments  increase  by  2-Tr/Tn  through 
the  succession.  The  corresponding  values  of  w'  are 


for  i  =  l,  ...,  m.  Now  a  ^-circuit  round  a,  that  is,  a  /-circuit  round  its 
origin,  increases  the  argument  of  z'  by  2?r  ;  hence  after  such  a  circuit  we 

1_        27Tt  !_ 

have  the  new  value  of  w{  as  ViZ/m  em,  that  is,  it  is  vi+1z'm  which  is  the  value 
of  w'i+l.  Hence  the  set  of  values  w\,  «/.,,...,  w'm  form  a  complete  set  of 
interchangeable  values  in  their  cyclical  succession  ;  all  the  m  values,  which 
are  equal  at  a,  form  a  single  cycle  and  the  point  is  a  branch-point. 

Next,  suppose  that  the  lowest  power  of  z  among  the  terms  independent 
of  w  is  z'  ,  where  I  >  1.     The  equation  now  has  the  form 
0  =  Az'  +  higher  powers  of  z' 
+  Bw'    +  higher  powers  of  w' 


Arsz'V 


r=l  s=l 

where  in  the  last  summation  r  and  s  are  not  zero  and  in  every  term  either 
(i),  r  is  equal  to  or  greater  than  I  or  (ii),  s  is  equal  to  or  greater  than  m 
or  (iii),  both  (i)  and  (ii)  are  satisfied.  As  only  terms  of  the  lowest  orders 


170 


BRANCH-POINTS 


[97. 


need  be  retained  for  the  present  purpose,  which  is  the  derivation  of  the  first 
term  of  w'  in  its  expansion  in  powers  of  z',  we  may  use  the  foregoing  equation 
in  the  form 


,        l-lm-l 

A/  +  2    2  A, 

r=l s=l 


,r    ,s    ,     -p.     ,m        _ 

jf  w   +  Bw     =  0. 


To  obtain  this  first  term  we  proceed  in  a  manner  similar  to  that  in  §  96  *. 
Points  A0,...,  Am  are  taken  in  a  plane 
referred  to  rectangular  axes  having  as  co 
ordinates  0,  £;...;  s,  r;...;  m,  0  respectively. 
A  line  is  taken  through  Am  and  is  made  to 
turn  round  Am  from  the  position  AmO  until 
it  first  meets  one  of  the  other  points ;  then 
round  the  last  point  which  lies  in  this 
direction,  say  round  Aj,  until  it  first  meets 
another ;  and  so  on. 

Any  line  through  At  (the  point  si}  rt)  is 
of  the  form 

y  -  Ti  =  -  \  (x  -  s^. 

The  intercept  on  the  axis  of  /-indices  is  \Si  +  Ti,  that  is,  the  order  of  the 
term  involving  Ars  for  a  substitution  w'  oc  /  .  The  perpendicular  from  the 
origin  for  a  line  through  AI  and  Aj  is  less  than  for  any  parallel  line  through 
other  points  with  the  same  inclination ;  and,  as  this  perpendicular  is 


Fig.  21. 


it  follows  that,  for  the  particular  substitution  w'  oc  z'  ,  the  terms  corresponding 
to  the  points  lying  on  the  line  with  coordinate  X  are  the  terms  of  lowest 
order  and  consequently  they  are  the  terms  which  give  the  initial  terms  for 
the  associated  set  of  quantities  w'. 

Evidently,  from  the  indices  retained  in  the  equation,  the  quantities  X 
for  the  various  pieces  of  the  broken  line  from  Am  to  A0  are  positive  and 
finite. 

Consider  the  first  piece,  from  Am  to  Aj  say ;  then  taking  the  value  of  X  for 

that  piece  as  fa,  so  that  we  write  v^z'*1  as  the  first  term  of  w',  we  have  as  the 
set  of  terms  involving  the  lowest  indices 

J?      /"*    i     ^  ^     A          fl*      fi    I       A  fl*i  ,  ^J 

Sj  being  the  smallest  value  of  s  retained ;  and  then 


so  that 


/*!   = 


m  —  s 


*  Reference  in  this  connection  may  be  made  to  Chrystal's  Algebra,   ch.  xxx.,  with  great 
advantage,  as  well  as  the  authorities  quoted  on  p.  168,  note. 


GROUPING   OF   BRANCHES  171 

Let  p/q  be  the  equivalent  value  of  ^  as  the  fraction  in  its  lowest  terms  ;  and 

p 

write  /  =  (?.     Then  w'  =  vlz'i  =  vtf  ;  all  the  terms  except  the  above  group 

are  of  order  >  mp  and  therefore  the  equation  leads  after  division  by  %mPtfi  to 

Bv^-'i  +  ^Aravf-*i  +  Arfj  =  0, 

an  equation  which  determines  m  —  Sj  values  for  vl,  and  therefore  the  initial 
terms  of  m  —  Sj  of  the  w-branches. 

Consider  now  the  second  piece,  from  Aj  to  At  say  ;  then  taking  the  value 

of  A,  for  that  piece  as  fa,  so  that  we  write  v.2z'^  as  the  first  term  of  w',  we 
have  as  the  set  of  terms  involving  the  lowest  indices  for  this  value  of  /*2 

A  fri     /s.-        xr"O    A         iv     Is  fl"i     tsi 

Arfz'  Jw  '  +  E&A.rjt  w'  +  Ar.sz'  *w  \ 
where  S{  is  the  smallest  value  of  s  retained.     Then 

Sjfr  +  Tj  =  tyig  +  r 

Proceeding  exactly  as  before,  we  find 


as  the  equation  determining  Sj-Si  values  for  v2  and  therefore  the  initial 
terms  of  Sj  —  st  of  the  w-branches. 

And  so  on,  until  all  the  pieces  of  the  line  are  used  ;  the  initial  terms  of 
all  the  w-branches  are  thus  far  determined  in  groups  connected  with  the 
various  pieces  of  the  line  A^Ai^.A,.  By  means  of  these  initial  terms, 
the  m-branches  can  be  arranged  for  their  interchanges,  by  the  description  of 
a  small  circuit  round  the  branch-point,  according  to  the  following  theorem  :— 

Each  group  can  be  resolved  into  systems,  the  members  of  each  of  which  are 
cyclically  interchangeable. 

It  will  be  sufficient  to  prove  this  theorem  for  a  single  group,  say  the 
group  determined  by  the  first  piece  of  broken  line:  the  argument  is 
general. 

Since  -  is  the  equivalent  of  —  ^—  and  of     T}  .   and  since  s,  <  s,  we  have 
V  m  —  s  m  —  Sj 

m-s  =  kq,         m-Sj^kjq,         kj>k; 
and  then  the  equation  which  determines  ^  is 

Sv&v  +  2^r,,Vl  <*,-*>  1  4  ArjSj  =  0, 

that  is,  an  equation  of  degree  k}  in  vj  as  its  variable.  Let  U  be  any  root  of 
it  ;  then  the  corresponding  values  of  vl  are  the  values  of  U*.  Suppose  these 
q  values  to  be  arranged  so  that  the  arguments  increase  by  27r^,  which  is 

possible,  because  p  is  prime  to  q.  Then  the  q  values  of  w'  being  the  values 
of  v^Vi  are 

P.  P  P 


172  GROUPING   OF   BRANCHES  [97. 

where  vla  is  that  value  of  Ifi  which  has  —  —  for  its  argument.     A  circuit 

round  the  /-origin  evidently  increases  the  argument  of  any  one  of  these 
w'-values  by  Zrrp/q,  that  is,  it  changes  it  into  the  value  next  in  the  succession; 
and  so  the  set  of  q  values  is  a  system  the  members  of  which  are  cyclically 
interchangeable. 

This  holds  for  each  value  of  U  derived  from  the  above  equation  ;  so  that 
the  whole  set  of  m  —  Sj  branches  are  resolved  into  kj  systems,  each  containing 
q  members  with  the  assigned  properties. 

It  is  assumed  that  the  above  equation  of  order  kj  in  vj  has  its  roots  unequal. 
If,  however,  it  should  have  equal  roots,  it  must  be  discussed  ab  initio  by  a 
method  similar  to  that  for  the  general  equation;  as  the  order  kj  (being  a 
factor  of  m  —  Sj)  is  less  than  m,  the  discussion  will  be  shorter  and  simpler, 
and  will  ultimately  depend  on  equations  with  unequal  roots  as  in  the  case 
above  supposed. 

It  may  happen  that  some  of  the  quantities  /j,  are  integers,  so  that  the 
corresponding  integers  q  are  unity  :  a  number  of  the  branches  would  then  be 
uniform  at  the  point. 

It  thus  appears  that  z  =  a  is  a  branch-point  and  that,  under  the  present 
circumstances,  the  m  branches  of  the  function  can  be  arranged  in  systems, 
the  members  of  each  one  of  which  are  cyclically  interchangeable. 

Lastly,  it  has  been  tacitly  assumed  in  what  precedes  that  the  common 
value  of  w  for  the  branch-point  is  finite.  If  it  be  infinite,  this  infinite  value 
can,  by  §  95,  arise  only  out  of  singularities  of  the  coefficients  of  the  equation  : 
and  there  is  therefore  a  reversion  to  the  discussion  of  §§  95,  96.  The  dis 
tribution  of  the  various  branches  into  cyclical  systems  can  be  carried  out 
exactly  as  above. 

Another  method  of  proceeding  for  these  infinities  would  be  to  take 
ww'  =  \,  z=  c  +  z'  ;  but  this  method  has  no  substantial  advantage  over  the 
earlier  one  and,  indeed,  it  is  easy  to  see  that  there  is  no  substantial 
difference  between  them. 

Ex.  1.     As  an  example,  consider  the  function  determined  by  the  equation 


The  equation  determining  the  values  of  z  which  give  equal  roots  for  w  is 

82  (2  -1)2  =  4(3  -I)3 
so  that  the  values  are  z=l  (repeated)  and  z=  —  1. 

When  z=l,  then  w=0,  occurring  thrice;  and,  if  2  =  1+2'  then 

8W/3W, 

that  is,  w'^^z13. 

The  three  values  are  branches  of  one  system  in  cyclical  order  for  a  circuit  round  z=\. 


97.]  EXAMPLES  173 

When  z  =  —  1,  the  equation  for  w  is 


that  is,  (w 

so  that  w=\  or  w=  —  £,  occurring  twice. 

For  the   former  of  these  we    easily   find   that,   for  s=  —  l-\-z',   the   value   of    w    is 
l-hfs'-f  ......  ,    an    isolated    branch    as    is    to    be    expected,    for    the    value    1    is    not 

repeated. 

For  the  latter  we  take  w——  \  +  w'  and  find 

so  that  the  two  branches  are 


and  they  are  cyclically  interchangeable  for  a  small  circuit  round  z=  -  1. 

These  are  the  finite  values  of  w  at  branch-points.     For  the  infinities  of  w,  which  may 
arise  in   connection  with  the  singularities  of  the  coefficients,  we  take   the  zeros  of  the 
coefficient  of  the  highest  power  of  w  in  the  integral  equation,  viz.,  2  =  0,  which  is  thus  the 
only  infinity  of  w.     To  find  its   order  we   take  w=z~nf  (z)—yz~n  +  ......  ,   where   y   is  a 

constant  and  f  (z)  is  finite  for  2  =  0;  and  then  we  have 

8zl~3n 


J.    "~ 

Thus  l-3n=-n, 

provided  both  of  them  be  negative;  the  equality  gives  n  =  \  and  satisfies  the  condition. 
And  8y3=  -  3y.  Of  these  values  one  is  zero,  and  gives  a  branch  of  the  function  without 
an  infinity;  the  other  two  are  ±^V-f  and  they  give  the  initial  term  of  the  two 
branches  of  w,  which  have  an  infinity  of  order  -^  at  the  origin  and  are  cyclically 
interchangeable  for  a  small  circuit  round  it.  The  three  values  of  w  for  infinitesimal 
values  of  z  are 

3  .  _i     1       7        /3  .  l      4          275 
"  -  81  '-1944 


3  •  - 


M  +  —    /?&*— 1*4. 215     /3-f_jL2_ 

6     18  V  8          81        1944  V  8          729 z      


_  _i     A        As 
w3--g  +  gj2+— 2  + 

The  first  two  of  these  form  the  system  for  the  branch-point  at  the  origin,  which  is  neither 
an  infinity  nor  a  critical  point  for  the  third  branch  of  the  function. 

Ex.  2.     Obtain  the  branch-points  of  the  functions  which  are  defined  by  the  following 
equations,  and  determine  the  cyclical  systems  at  the  branch-points  : 
(i)       w* 
(ii)       w 
(iii)     w 
(iv)     iff 


44 
(v)      vfi  -  (1  -  a2)  104  _  _  Z2  (!  _  22)4  =  0-  (Briot  and  Bouquet.) 

Also  discuss  the  branches,  in  the  vicinity  of  2  =  0  and  of  2=00,  of  the  functions  defined 
by  the  following  equations  : 

(vi)     aw7  +  bu£z + cutz*  +  dwW  +  ewz1  +fz9  +  gv£ + hw*£ + kzw  =  0 ; 
(vii)     wmzn+wn+zm  =  Q. 


174  SIMPLE   BRANCH-POINTS  [98. 

98.  There  is  one  case  of  considerable  importance  which,  though  limited 
in  character,  is  made  the  basis  of  Clebsch  and  Gordan's  investigations*  in  the 
theory  of  Abelian  functions  —  the  results  being,  of  course,  restricted  by  the 
initial  limitations.  It  is  assumed  that  all  the  branch-points  are  simple,  that 
is,  are  such  that  only  one  pair  of  branches  of  w  are  interchanged  by  a  circuit 
of  the  variable  round  the  point  ;  and  it  is  assumed  that  the  equation  /=  0  is 
algebraical  not  merely  in  w  but  also  in  z.  The  equation  f  =  0  can  then  be 
regarded  as  the  generalised  form  of  the  equation  of  a  curve  of  the  nth  order, 
the  generalisation  consisting  in  replacing  the  usual  coordinates  by  complex 
variables;  and  it  is  further  assumed,  in  order  to  simplify  the  analysis,  that  all 
the  multiple  points  on  the  curve  are  (real  or  imaginary)  double-points.  But, 
even  with  the  limitations,  the  results  are  of  great  value  :  and  it  is  therefore 
desirable  to  establish  the  results  that  belong  to  the  present  section  of  the 
subject. 

We  assume,  therefore,  that  the  branch-points  are  such  that  only  one 
pair  of  branches  of  w  are  interchanged  by  a  small  closed  circuit  round  any 
one  of  the  points.  The  branch-points  are  among  the  values  of  z  determined 
by  the  equations 

z)     A 
> 


When  /=0  has  the  most  general  form  consistent  with  the  assigned 
limitations,  f  (w,  z)  is  of  the  ?ith  degree  in  z  ;  the  values  of  z  are  determined 
by  the  eliminant  of  the  two  equations  which  is  of  degree  n(n  —  1),  and  there 
are,  therefore,  n(n  —  Y)  values  of  z  which  must  be  examined. 

First,  suppose   that    J  \,  '  —  '  does   not   vanish  for   a   value   of  z,  thus 

oz 

obtained,  and  the  corresponding  value  of  w  :  then  we  have  the  first  case 
in  the  preceding  investigation.  And,  on  the  hypothesis  adopted  in  the 
present  instance,  m  =  2  ;  so  that  each  such  point  z  is  a  branch-point. 

Next,  suppose  that  —  ^  -  vanishes  for  some  of  the  n(n  —  1)  values  of  z  ; 

the  value  of  m  is  still  2,  owing  to  the  hypothesis.     The  case  will  now  be  still 

d'2f  (w  z} 
further  limited  by  assuming  that        ^  .2        does  not  vanish  for  the  value  of  z 

and  the  corresponding  value  of  w  ;  and  thus  in  the  vicinity  of  z  =  a,  w  =  a  we 
have  an  equation 

0  =  Az-  +  2Bz'w'  +  Cw'2  -f  terms  of  the  third  degree  +  ......  , 

where  A,  B,  C  are  the  values  of  ^  ,   =-£-  ,    «~  f°r  z  —  a>  w=a. 

oz1     dzdw     ow2 

If  B2     AC,  this  equation  leads  to  the  solution 

C'w  +  Bz  oc  uniform  function  of  z. 

*  Clebsch  und  Gordan,  Theorie  der  AbeVschen  Functionen,  (Leipzig,  Teubner,  1866). 


98.]  SIMPLE    BRANCH-POINTS  175 

The  point  z  =  a,  w  =  a  is  not  a  branch-point ;  the  values  of  w,  equal  at  the 
point,  are  functionally  distinct.  Moreover,  such  a  point  z  occurs  doubly  in 
the  eliminant;  so  that,  if  there  be  B  such  points,  they  account  for  28  in 
the  eliminant  of  degree  n  (n  —  1) ;  and  therefore,  on  their  score,  the  number 
n  (n  —  1)  must  be  diminished  by  '28.  The  case  is,  reverting  to  the  genera 
lisation  of  the  geometry,  that  of  a  double  point  where  the  tangents  are 
not  coincident. 

If,  however,  B2  =  AC,  the  equation  leads  to  the  solution 

Cw'  +  Bz'  =  Lz'^  +  Mz'*  +  Nz'*  + 

The  point  z  =  a,  w  =  a  is  a  point  where  the  two  values  of  z  interchange. 
Now  such  a  point  z  occurs  triply  in  the  eliminant ;  so  that,  if  there  be  K 
such  points,  they  account  for  SK  of  the  degree  of  the  equation.  Each  of 
them  provides  only  one  branch-point,  and  the  aggregate  therefore  provides  K 
branch-points ;  hence,  in  counting  the  branch-points  of  this  type  as  derived 
through  the  eliminant,  its  degree  must  be  diminished  by  2/c.  The  case  is, 
reverting  to  the  generalisation  of  the  geometry,  that  of  a  double  point  (real 
or  imaginary)  where  the  tangents  are  coincident. 

It  is  assumed  that  all  the  n(n—  1)  points  z  are  accounted  for  under 
the  three  classes  considered.  Hence  the  number  of  branch-points  of  the 
equation  is 

£l  =  n  (n  -  1)  -  28  -  2«, 

where  n  is  the  degree  of  the  equation,  B  is  the  number  of  double  points 
(in  the  generalised  geometrical  sense)  at  which  tangents  to  the  curve  do  not 
coincide,  and  K  is  the  number  of  double  points  at  which  tangents  to  the 
curve  do  coincide. 

And  at  each  of  these  branch-points,  II  in  number,  two  branches  of  the 
function  are  equal  and,  for  a  small  circuit  round  it,  interchange. 

99.  The  following  theorem  is  a  combined  converse  of  many  of  the 
theorems  which  have  been  proved : 

A  function  w,  which  has  n  (and  only  ?/)  values  for  each  value  of  z,  and 
which  has  a  finite  number  of  infinities  and  of  branch-points  in  any  part  of  the 
plane,  is  a  root  of  an  equation  in  w  of  degree  n,  the  coefficients  of  which  are 
uniform  functions  of  z  in  that  part  of  the  plane. 

We  shall  first  prove  that  every  integral  symmetric  function  of  the  n 
values  is  a  uniform  function  in  the  part  of  the  plane  under  consideration. 

n 

Let  Sk  denote  2,  w£,  where  k  is  a  positive  integer.     At  an  ordinary  point 

i-\ 

of  the  plane,  Sk  is  evidently  a  one-valued  function  and  that  value  is  finite ; 
Sk  is  continuous ;  and  therefore  the  function  Sk  is  uniform  in  the  immediate 
vicinity  of  an  ordinary  point  of  the  plane. 


176  FUNCTIONS   POSSESSING  [99. 

For  a  point  a,  which  is  a  branch-point  of  the  function  w,  we  know  that 
the  branches  can  be  arranged  in  cyclical  systems.  Let  w1,...,  w^  be  such  a 
system.  Then  these  branches  interchange  in  cyclical  order  for  a  description 
of  a  small  circuit  round  a  ;  and,  if  z  —  a  =  Z*,  it  is  known  (§  93)  that,  in  the 
vicinity  of  Z  =  0,  a  branch  w  is  a,  uniform  function  of  Z,  say 


Therefore  wk  =  Gk       )  +  Pk  (Z) 

\£il 

in  the  vicinity  of  Z  =  0  ;  say 

w*  =  Ak  +   2Bk>mZ-™  +    2  Ck>mZ™. 

m=l  m=l 

Now  the  other  branches  of  the  function  which  are  equal  at  a  are  derivable 
from  any  one  of  them  by  taking  the  successive  values  which  that  one 
acquires  as  the  variable  describes  successive  circuits  round  a.  A  circuit 
of  w  round  a  changes  the  argument  of  z  —  a,  by  27r.  and  therefore  gives  Z 
reproduced  but  multiplied  by  a  factor  which  is  a  primitive  /xth  root  of  unity, 
say  by  a  factor  a  ;  a  second  circuit  will  reproduce  Z  with  a  factor  a2  ;  and  so 
on.  Hence 

wf  =  Ak+2  Bk>m  a—  Z-™  +ZCk>m  a-  #» 


wrk  =  Ak+2  Bk>m a-™ Z~m  +  2  Ck,m  a™ Zm, 

m=l  »»=! 


and  therefore 
I* 

wrk  =  pAk  +  2  Bkm    -        +  ar    +  cr     +  .  .  .  +  cr't 

r=l  m  =  1 

+  2  flto*  Zm  (1  +  «m  +  a2"*  +  •  •  •  +  a""*-™). 

OT  =  1 

Now,  since  a  is  a  primitive  /*th  root  of  unity, 

1  +as  +  «2S+  ...  +  as('x-1) 

is  zero  for  all  integral  values  of  s  which  are  not  integral  multiples  of  p,,  and  it 
is  yu,  for  those  values  of  s  which  are  integral  values  of  jj,  ;  hence 

-  £ 


B'k>  i(z  -  a)"1  +  B'k^(z  -  a)~2  +  B'kt3  (z  -  a) 


. 
Hence  the  point  z  =  a  may  be  a  singularity  of  2  wrk  but  it  is  not  a  branch- 

r=l 


99.]  A    FINITE    NUMBER   OF   BRANCHES  177 

point  of  the  function  ;  and  therefore  in  the  immediate  vicinity  of  z  —  a  the 

*i 
quantity  X  wrk  is  a  uniform  function. 


r=l 


The  point  a  is  an  essential  singularity  of  this  uniform  function,  if  the 
order  of  the  infinity  of  w  at  a  be  infinite  :  it  is  an  accidental  singularity,  if 
that  order  be  a  finite  integer. 

This  result  is  evidently  valid  for  all  the  cyclical  systems  at  a,  as  well  as 
for  the  individual  branches  which  may  happen  to  be  one-valued  at  a.  Hence 

(U. 

Sk,  being  the  sum  of  sums  of  the  form  2)  wrk  each  of  which  is  a  uniform 

r=l 

function  of  z  in  the  vicinity  of  a,  is  itself  a  uniform  function  of  z  in  that 
vicinity.  Also  a  is  an  essential  singularity  of  Sk,  if  the  order  of  the  infinity  at 
z  =  a  for  any  one  of  the  branches  of  w  be  infinite  ;  and  it  is  an  accidental 
singularity  of  Sk>  if  the  order  of  the  infinity  at  z  =  a  for  all  the  branches  of  w 
be  finite.  Lastly,  it  is  an  ordinary  point  of  Sk,  if  there  be  no  branch  of  w  for 
which  it  is  an  infinity.  Similarly  for  each  of  the  branch-points. 

Again,  let  c  be  a  regular  singularity  of  any  one  (or  more)  of  the  branches 
of  w  ;  then  c  is  a  regular  singularity  of  every  power  of  each  of  those  branches, 
the  singularities  being  simultaneously  accidental  or  simultaneously  essential. 
Hence  c  is  a  singularity  of  8k  :  and  therefore  in  the  vicinity  of  c,  $&  is  a 
uniform  function,  having  c  for  an  accidental  singularity  if  it  be  so  for  each  of 
the  branches  w  affected  by  it,  and  having  c  for  an  essential  singularity  if  it  be 
so  for  any  one  of  the  branches  w. 

It  thus  appears  that  in  the  part  of  the  plane  under  consideration  the 
function  8k  is  one-valued  ;  and  it  is  continuous  and  finite,  except  at  certain 
isolated  points  each  of  which  is  a  singularity.  It  is  therefore  a  uniform 
function  in  that  part  of  the  plane  ;  and  the  singularity  of  the  function  at  any 
point  is  essential,  if  the  order  of  the  infinity  for  any  one  of  the  branches  w  at 
that  point  be  infinite,  but  it  is  accidental,  if  the  order  of  the  infinity  for  all  the 
branches  w  there  be  finite.  And  the  number  of  these  singularities  is  finite, 
being  not  greater  than  the  combined  number  of  the  infinities  of  the  function 
w,  whether  regular  singularities  or  branch-points. 

Since  the  sums  of  the  kth  powers  for  all  positive  values  of  the  integer  k 
are  uniform  functions  and  since  any  integral  symmetric  function  of  the  n 
values  is  a  rational  integral  algebraical  function  of  the  sums  of  the  powers,  it 
follows  that  any  integral  symmetric  function  of  the  n  values  is  a  uniform 
function  of  z  in  the  part  of  the  plane  under  consideration  ;  and  every  infinity 
of  a  branch  w  leads  to  a  singularity  of  the  symmetric  function,  which  is 
essential  or  accidental  according  as  the  orders  of  infinity  of  the  various 
branches  are  not  all  finite  or  are  all  finite. 

F.  12 


178  FUNCTIONS   POSSESSING  [99. 

Since  w  has  n  (and  only  n)  values  wlt...  ,wn  for  each  value  of  z,  the 
equation  which  determines  w  is 

(W  -  Wj)  (W-W2)  ...  (W-  Wn)  =  0. 

The  coefficients  of  the  various  powers  of  w  are  symmetric  functions  of  the 
branches  wl , . . . ,  wn;  and  therefore  they  are  uniform  functions  of  z  in  the 
part  of  the  plane  under  consideration.  They  possess  a  finite  number  of 
singularities,  which  are  accidental  or  essential  according  to  the  character  of 
the  infinities  of  the  branches  at  the  same  points. 

COROLLARY.  If  all  the  infinities  of  the  branches  in  the  finite  part  of  the 
whole  plane  be  of  finite  order,  then  the  finite  singularities  of  all  the  coefficients 
of  the  powers  of  w  in  the  equation  satisfied  by  w  are  all  accidental ;  and  the 
coefficients  themselves  then  take  the  form  of  a  quotient  of  an  integral  uniform 
function  (which  may  be  either  transcendental  or  algebraical,  in  the  sense  of 
§  47)  by  another  function  of  a  similar  character. 

If  z  =  oc  be  an  essential  singularity  for  at  least  one  of  the  coefficients, 
through  being  an  infinity  of  unlimited  order  for  a  branch  of  w,  then  one 
or  both  of  the  functions  in  the  quotient-form  of  one  at  least  of  the  coefficients 
must  be  transcendental. 

If  z  =  oo  be  an  accidental  singularity  or  an  ordinary  point  for  all  the 
coefficients,  through  being  either  an  infinity  of  finite  order  or  an  ordinary 
point  for  the  branches  of  w,  then  all  the  functions  which  occur  in  all  the 
coefficients  are  rational,  algebraical  expressions.  When  the  equation  is 
multiplied  throughout  by  the  least  common  multiple  of  the  denominators 
of  the  coefficients,  it  takes  the  form 

wnh0  (z)  +  wn~*  A,  (z)  +  . . .  +  w  hn_,  (z}  +  hn  (z)  =  0, 

where  the  functions  h0(z),  h^(z\ ...,  hn(z)  are  rational,  integral,  algebraical 
functions  of  z,  in  the  sense  of  §  47. 

A  knowledge  of  the  number  of  infinities  of  w  gives  an  upper  limit  of  the 
degree  of  the  equation  in  z  in  the  last  form.  Thus,  let  at  be  a  regular 
singularity  of  the  function  ;  and  let  Oi,  fa,  ji, ...  be  the  orders  of  the  infinities 
of  the  branches  at  at- ;  then 

w^w-i ...  wn(z  —  at )A', 
where  \  denotes  Oi  + fti +  %  +  ...,  is  finite  (but  not  zero)  for  z  =  at. 

Let  Ci  be  a  branch-point,  which  is  an  infinity;  and  let  p,  branches  w  form  a 

ft 
system  for  ct-,  such  that  w(z  —  Cf)^  is  finite  (but  not  zero)  at  the  point;  then 

w:w2 ...  Wp  (z  —  Q)  ' 
is  finite  (but  not  zero)  at  the  point,  and  therefore  also 


99.]  A   FINITE   NUMBER  OF   BRANCHES  179 

is  finite,  where  Qit  (/>;,  ^i,  ...  are  numbers  belonging  to  the  various  systems; 
or,  if  ei  denote  0;  +  $f  +  tyi  +  .  .  .  ,  then 

Wl...Wn(z-  Ci)6i 

is  finite  for  z  =  C;.     Similarly  for  other  symmetric  functions  of  w. 

Hence,  if  «j,  a2,  ...  be  the  regular  singularities  with  numbers  X1;  X2,  ... 
defined  as  above,  and  if  c^  c2,  ...  be  the  branch  -points,  that  are  also  infinities, 
with  numbers  e1;  e2,  ...  defined  as  above,  then  the  product 

(w-Wj)  ......  (w-wn)  n  0-af)A<  n  0-Ci)e< 

i=l  1=1 

is  finite  at  all  the  points  ai  and  at  all  the  points  c;.  The  points  a  and  the 
points  c  are  the  only  points  in  the  finite  part  of  the  plane  that  can  make  the 
product  infinite  :  hence  it  is  finite  everywhere  in  the  finite  part  of  the  plane, 
and  it  is  therefore  an  integral  function  of  z. 

Lastly,  let  p  be  the  number  for  z  =  oo  corresponding  to  \i  for  af  or  to  e^ 
for  C;,  so  that  for  the  coefficient  of  any  power  of  w  in  (w  —  w^)  ...(w  —  wn)  the 
greatest  difference  in  degree  between  the  numerator  and  the  denominator  is 
p  in  favour  of  the  excess  of  the  former. 

Then  the  preceding  product  is  of  order 


which  is  therefore  the  order  of  the  equation  in  z  when  it  is  expressed  in  a 
holomorphic  form. 


12—2 


CHAPTER   IX. 

PERIODS  OF  DEFINITE  INTEGRALS,  AND  PERIODIC  FUNCTIONS  IN  GENERAL. 

100.  INSTANCES  have  already  occurred  in  which  the  value  of  a  function 
of  z  is  not  dependent  solely  upon  the  value  of  z  but  depends  also  on  the 
course  of  variation  by  which  z  obtains  that  value ;  for  example,  integrals  of 
uniform  functions,  and  multiform  functions.  And  it  may  be  expected  that, 
a  fortiori,  the  value  of  an  integral  connected  with  a  multiform  function  will 
depend  upon  the  course  of  variation  of  the  variable  z.  Now  as  integrals 
which  arise  in  this  way  through  multiform  functions  and,  generally,  integrals 
connected  with  differential  equations  are  a  fruitful  source  of  new  functions, 
it  is  desirable  that  the  effects  on  the  value  of  an  integral  caused  by  variations 
of  a  £-path  be  assigned  so  that,  within  the  limits  of  algebraic  possibility,  the 
expression  of  the  integral  may  be  made  completely  determinate. 

There  are  two  methods  which,  more  easily  than  others,  secure  this  result ; 
one  of  them  is  substantially  due  to  Cauchy,  the  other  to  Riemann. 

The  consideration  of  Riemann's  method,  both  for  multiform  functions  and 
for  integrals  of  such  functions,  will  be  undertaken  later,  in  Chapters  XV., 
XVI.  Cauchy's  method  has  already  been  used  in  preceding  sections  relating 
to  uniform  functions,  and  it  can  be  extended  to  multiform  functions.  Its 
characteristic  feature  is  the  isolation  of  critical  points,  whether  regular 
singularities  or  branch-points,  by  means  of  small  curves  each  containing  one 
and  only  one  critical  point. 

Over  the  rest  of  the  plane  the  variable  z  ranges  freely  and,  under  certain 
conditions,  any  path  of  variation  of  z  from  one  point  to  another  can,  as  will 
be  proved  immediately,  be  deformed  without  causing  any  change  in  the 
value  of  the  integral,  provided  that  the  path  does  not  meet  any  of  the  small 
curves  in  the  course  of  the  deformation.  Further,  from  a  knowledge  of  the 
relation  of  any  point  thus  isolated  to  the  function,  it  is  possible  to  calculate 
the  change  caused  by  a  deformation  of  the  £-path  over  such  a  point;  and 
thus,  for  defined  deformations,  the  value  of  the  integral  can  be  assigned 
precisely. 


100.]  INTEGRAL   OF   A   BRANCH  181 

The  properties  proved  in  Chapter  II.  are  useful  in  the  consideration  of 
the  integrals  of  uniform  functions ;  it  is  now  necessary  to  establish  the 
propositions  which  give  the  effects  of  deformation  of  path  on  the  integrals 
of  multiform  function.  The  most  important  of  these  propositions  is  the 
following : — 

fb 
If  w  be  a  multiform  function,  the  value  of  I   wdz,  taken  between  two 

J  a 

ordinary  points,  is  unaltered  for  a  deformation  of  the  path,  provided  that  the 
initial  branch  of  w  be  the  same  and  that  no  branch-point  or  infinity  be  crossed 
in  the  deformation. 

Consider  two  paths  acb,  adb,  (fig.  16,  p.  152),  satisfying  the  conditions 
specified  in  the  proposition.  Then  in  the  area  between  them  the  branch  w 
has  no  infinity  and  no  point  of  discontinuity ;  and  there  is  no  branch-point 
in  that  area.  Hence,  by  §  90,  Corollary  VI.,  the  branch  w  is  a  uniform 
monogenic  function  for  that  area;  it  is  continuous  and  finite  everywhere 
within  it  and,  by  the  same  Corollary,  we  may  treat  w  as  a  uniform,  mono 
genic,  finite  and  continuous  function.  Hence,  by  §  17,  we  have 

rb  ra 

(c)  I    wdz  +  (d)      wdz  =  0, 

J  a  J  b 

the  first  integral  being  taken  along  acb  and  the  second  along  bda;  and 
therefore 

rb  ra  rb 

(c)      wdz  =  —  (d}\    wdz  =  (d)  \    wdz, 

Jo,  J  b  J  a 

shewing  that  the  values  of  the  integral  along  the  two  paths  are  the  same 
under  the  specified  conditions. 

It  is  evident  that,  if  some  critical  point  be  crossed  in  the  deformation, 
the  branch  w  cannot  be  declared  uniform  and  finite  in  the  area  and  the 
theorem  of  §  17  cannot  then  be  applied. 

COROLLARY  I.  The  integral  round  a  closed  curve  containing  no  critical 
point  is  zero. 

COROLLARY  II.  A  curve  round  a  branch-point,  containing  no  other 
critical  point  of  the  function,  can  be  deformed  into  a  loop 
without  altering  the  value  of  fwdz ;  for  the  deformation 
satisfies  the  condition  of  the  proposition.  Hence,  when 
the  value  of  the  integral  for  the  loop  is  known,  the 
value  of  the  integral  is  known  for  the  curve. 

COROLLARY  III.  From  the  proposition  it  is  possible 
to  infer  conditions,  under  which  the  integral  fwdz  round 
the  whole  of  any  curve  remains  unchanged,  when  the  whole 
curve  is  deformed,  without  leaving  an  infinitesimal  arc 
common  as  in  Corollary  II. 


182  INTEGRATION  [100. 

Let  GDC',  ABA'  be  the  curves:  join  two  consecutive  points  A  A'  to  two 
consecutive  points  (7(7.  Then  if  the  area  CABA'C'DG 
enclose  no  critical  point  of  the  function  w,  the  value  of 
jwdz  along  CDC'  is  by  the  proposition  the  same  as  its 
value  along  CABA'C'.  The  latter  is  made  up  of  the 
value  along  CA,  the  value  along  ABA',  and  the  value 
along  AC',  say 

rA  r  rC' 

I    wdz  +  I    wdz  +        w'dz,  v. 

Jc  JB  JA  Ǥ. 

where  w'  is  the  changed  value  of  w  consequent  on  the  description  of  a  simple 
curve  reducible  to  B  (§  90,  Cor.  II.). 

Now  since  w  is  finite  everywhere,  the  difference  between  the  values  of  w 
at  A  and  at  A'  consequent  on  the  description  of  ABA  is  finite  :  hence  as 
A  A  is  infinitesimal  the  value  of  jwdz  necessary  to  complete  the  value  for 
the  whole  curve  B  is  infinitesimal  and  therefore  the  complete  value  can  be 

taken  as  the  foregoing  integral       wdz.     Similarly  for  the  complete  value 

J  B 

along  the  curve  D :  and  therefore  the  difference  of  the  integrals  round  B  and 
round  D  is 

rA  rC' 

I    wdz  +  I     w'dz, 

J  C  J  A' 

rA 

say  (w  —  w')  dz. 

J  c 

In  general  this  integral  is  not  zero,  so  that  the  values  of  the  integral 
round  B  and  round  D  are  not  equal  to  one  another :  and  therefore  the  curve 
D  cannot  be  deformed  into  the  curve  B  without  affecting  the  value  of  jwdz 
round  the  whole  curve,  even  when  the  deformation  does  not  cause  the  curve 
to  pass  over  a  critical  point  of  the  function. 

But  in  special  cases  it  may  vanish.  The  most  important  and,  as  a 
matter  of  fact,  the  one  of  most  frequent  occurrence  is  that  in  which  the 
description  of  the  curve  B  restores  at  A'  the  initial  value  of  w  at  A.  It 
easily  follows,  by  the  use  of  §  90,  Cor.  II.,  that  the  description  of  D  (as 
suming  that  the  area  between  B  and  D  includes  no  critical  point)  restores 
at  C'  the  initial  value  of  w  at  (7.  In  such  a  case,  w  =  w'  for  corresponding 
points  on  AC  and  A'C',  and  the  integral,  which  expresses  the  difference,  is 
zero:  the  value  of  the  integral  for  the  curve  B  is  then  the  same  as  that  for  D. 
Hence  we  have  the  proposition : — 

If  a  curve  be  such  that  the  description  of  it  by  the  independent  variable 
restores  the  initial  value  of  a  multiform  function  w,  then  the  value  of  jwdz 
taken  round  the  curve  is  unaltered  when  the  curve  is  deformed  into  any  other 
curve,  provided  that  no  branch-point  or  point  of  discontinuity  of  w  is  crossed 
in  the  course  of  deformation. 


100.]  OF   MULTIFORM    FUNCTIONS  183 

This  is  the  generalisation  of  the  proposition  of  §  19  which  has  thus  far 
been  used  only  for  uniform  functions. 

Note.  Two  particular  cases,  which  are  very  simple,  may  be  mentioned 
here  :  special  examples  will  be  given  immediately. 

The  first  is  that  in  which  the  curve  B,  and  therefore  also  D,  encloses 
no  branch-point  or  infinity;  the  initial  value  of  w  is  restored  after  a 
description  of  either  curve,  and  it  is  easy  to  see  (by  reducing  B  to  a 
point,  as  may  be  done)  that  the  value  of  the  integral  is  zero. 

The  second  is  that  in  which  the  curve  encloses  more  than  one  branch 
point,  the  enclosed  branch-points  being  such  that  a  circuit  of  all  the  loops, 
into  which  (by  Corollary  V.,  §  90)  the  curve  can  be  deformed,  restores  the 
initial  branch  of  w.  This  case  is  of  especial  importance  when  w  is  two-valued  : 
the  curves  then  enclose  an  even  number  of  branch-points. 

101.  It  is  important  to  know  the  value  of  the  integral  of  a  multiform 
function  round  a  small  curve  enclosing  a  branch-point. 

Let  c  be  a  point  at  which  TO  branches  of  an  algebraical  function  are  equal 
and  interchange  in  a  single  cycle  ;  and  let  c,  if  an  infinity,  be  of  only  finite 
order,  say  k/m.  Then  in  the  vicinity  of  c,  any  of  the  branches  w  can  be 
expressed  in  the  form 

00  .« 

w=    2    gs(z-c)m, 

o  —       If 

o  —  —  K 

where  k  is  a  finite  integer. 

The  value  of  jwdz  taken  round  a  small  curve  enclosing  c  is  the  sum  of 
the  integrals 


the  value  of  which,  taken  once  round  the  curve  and  beginning  at  a  point  zly  is 


TO  +  S 

where  a  is  a  primitive  mth  root  of  unity,  provided  TO  +  s  is  not  zero.  If  then 
s  +  m  be  positive,  the  value  is  zero  in  the  limit  when  the  curve  is  infini 
tesimal  :  if  TO  +  s  be  negative,  the  value  is  oo  in  the  limit. 

But,  if  m  +  s  be  zero,  the  value  is  Z7rigs. 

Hence  we  have  the  proposition:  If,  in  the  vicinity  of  a  branch-point  c, 
where  m  branches  w  are  equal  to  one  another  and  interchange  cyclically,  the 
expression  of  one  of  the  branches  be 


184  MULTIPLICITY   OF   VALUE  [101. 

then  jwdz,  taken  once  round  a  small  curve  enclosing  c,  is  zero,  if  k<m;  is 
infinite,  if  k>  m ;  and  is  ^irig^ ,  if  k  =  m. 

It  is  easy  to  see  that,  if  the  integral  be  taken  m  times  round  the  small 
curve  enclosing  c,  then  the  value  of  the  integral  is  2m7rigm  when  k  is  greater 
than  in,  so  that  the  integral  vanishes  unless  there  be  a  term  involving  (z  —  c)"1 
in  the  expansion  of  a  branch  w  in  the  vicinity  of  the  point.  The  reason  that 
the  integral,  which  can  furnish  an  infinite  value  for  a  single  circuit,  ceases  to 

_* 

do  so  for  m  circuits,  is  that  the  quantity  (^  —  c)  m,  which  becomes  indefi 
nitely  great  in  the  limit,  is  multiplied  for  a  single  circuit  by  a*—  1,  for  a 
second  circuit  by  a2A  —  aA,  and  so  on,  and  for  the  mth  circuit  by  awA  —  a(w~1)A, 
the  sum  of  all  of  which  coefficients  is  zero. 

Ex.  The  integral  \{(z  -  a)  (z  -  b) ...  (z  -f)}~*  dz  taken  round  an  indefinitely  small  curve 
enclosing  a  is  zero,  provided  no  one  of  the  quantities  b, ... ,/  is  equal  to  a. 

102.  Some  illustrations  have  already  been  given  in  Chapter  II.,  but 
they  relate  solely  to  definite,  not  to  indefinite,  integrals  of  uniform 
functions.  The  whole  theory  will  not  be  considered  at  this  stage ;  we  shall 
merely  give  some  additional  illustrations,  which  will  shew  how  the  method 
can  be  applied  to  indefinite  integrals  of  uniform  functions  and  to  integrals 
of  multiform  functions,  and  which  will  also  form  a  simple  and  convenient 
introduction  to  the  theory  of  periodic  functions  of  a  single  variable. 

We  shall  first  consider  indefinite  integrals  of  uniform  functions. 


f  dz 
Ex.  1.     Consider  the  integral  I  — ,  and  denote*  it  by/ (z}. 


The  function  to  be  integrated  is  uniform,  and  it  has  an  accidental  singularity  of  the  first 
order  at  the  origin,  which  is  its  only  singularity.  The  value  of  \z~l  dz  taken  positively 
along  a  small  curve  round  the  origin,  say  round  a  circle  with  the  origin  as  centre,  is  2n-i  • 
but  the  value  of  the  integral  is  zero  when  taken  along  any  closed  curve  which  does  not 
include  the  origin. 

Taking  z  =  l  as  the  lower  limit  of  the  integral,  and  any  point  z  as  the  upper  limit,  we 
consider  the  possible  paths  from  1  to  z.  Any  path  from  1  to  z  can  be  deformed,  without 
crossing  the  origin,  into  a  path  which  circumscribes  the  origin  positively  some  number  of 
times,  say  m^,  and  negatively  some  number  of  times,  say  »i2,  all  in  any  order,  and  then  leads 
in  a  straight  line  from  1  to  z.  For  this  path  the  value  of  the  integral  is  equal  to 


I    —  , 
J  1  z 

that  is,  to  2mni+  I    —  , 

Ji  z 

where  m  is  an  integer,  and  in  the  last  integral  the  variation  of  z  is  along  a  straight 
line  from  1  to  z.     Let  the  last  integral  be  denoted  by  u  ;  then 


*  See  Chrystal,  ii,  pp.  266  —  272,  for  the  elementary  properties  of  the  function  and  its  inverse, 
when  the  variable  is  complex. 


102.]  OF    INTEGRALS  185 

and  therefore,  inverting  the  function  and  denoting/"1  by  <j>,  we  have 


Hence  the  general  integral  is  a  function  of  z  with  an  infinite  number  of  values  ;  and  z  is  a 
periodic  function  of  the  integral,  the  period  being  2n-z. 

Ex.  2.     Consider  the  function  /  -  -  ^  >  and  again  denote  it  by  /  (z). 

The  one-  valued  function  to  be  integrated  has  two  accidental  singularities  +  i,  each  of 
the  first  order.  The  value  of  the  integral  taken  positively  along  a  small  curve  round  i  is 
TT,  and  along  a  small  curve  round  —  i  is  —  n. 

We  take  the  origin  0  as  the  lower  limit  and  any  point  z  as  the  upper  limit.  Any  path 
from  0  to  z  can  be  deformed,  without  crossing  either  of  the  singularities  and  therefore 
without  changing  the  value  of  the  integral,  into 

(i)     any  numbers  of  positive  (ml5  w?2)  an(*  of  negative  (nz/,  m2')  circuits  round  i  and 
round  -i,  and 

(ii)    a  straight  line  from  0  to  z. 
Then  we  have 

-  TJ-)  +WIJJ  (  -  IT)  +  m.2'  {_(-„•)}+  /*  . 

J  o 

,     z 
=  nir+ 


where  ?i  is  an  integer  and  the  integral  on  the  right-hand  side  is  taken  along  a  straight  line 
from  0  to  z. 

Inverting  the  function  and  denoting/"1  by  tp,  we  have 


The  integral,  as  before,  is  a  function  of  z  with  an  infinite  number  of  values ;  and  z  is  a 
periodic  function  of  the  integral,  the  period  being  TT. 

103.  Before  passing  to  the  integrals  of  multiform  functions,  it  is  con 
venient  to  consider  the  method  in  which  Hermite*  discusses  the  multiplicity 
in  value  of  a  definite  integral  of  a  uniform  function. 

Taking  a  simple  case,  let       <£>  (X)  =  \ 

J  Q    1   +  Z 

and  introduce  a  new  variable  t  such  that  Z—zt\  then 

zdt 

When  the  path  of  t  is  assigned,  the  integral  is  definite,  finite  and  unique  in 
value  for  all  points  of  the  plane  except  for  those  for  which  1  +  zt  =  0 ;  and, 
according  to  the  path  of  variation  of  t  from  0  to  1,  there  will  be  a  0-curve 
which  is  a  curve  of  discontinuity  for  the  subject  of  integration.  Suppose  the 
path  of  t  to  be  the  straight  line  from  0  to  1 ;  then  the  curve  of  discontinuity 

*  Crelle,  t.  xci,  (1881),  pp.  62—77;  Cours  a  la  Faculte  des  Sciences,  46me  6d.  (1891),  pp. 
76—79,  154—164,  and  elsewhere. 


186  HERMITE'S  [103. 

is  the  axis  of  x  between  —  1  and  —  oo  .  In  this  curve  let  any  point  -  £  be 
taken  where  £  >  1 ;  and  consider  a  point  z1  —  -^  +  ie  and  a  point  z2  =  —  £  —  ie, 
respectively  on  the  positive  and  the  negative  sides  of  the  axis  of  x,  both 
being  ultimately  taken  as  infinitesimally  near  the  point  —  £.  Then 


dt=  ( 


Let  e  become  infinitesimal  ;  then,  when  t  is  infinite,  we  have 


tan 


for  e  is  positive  ;  and,  when  t  is  unity,  we  have 


tan"1  -----    =  —  |TT, 


for  £  is  >  1.     Hence  <£  (^)  —  <£  (^2) 

The  part  of  the  axis  of  x  from  -  1  to  -  oo  is  therefore  a  line  of  discon 
tinuity  in  value  of  <j>  (z),  such  that  there  is  a  sudden  change  in  passing  from 
one  edge  of  it  to  the  other.  If  the  plane  be  cut  along  this  line  so  that 
it  cannot  be  crossed  by  the  variable  which  may  not  pass  out  of  the  plane, 
then  the  integral  is  everywhere  finite  and  uniform  in  the  modified  surface. 
If  the  plane  be  not  cut  along  the  line,  it  is  evident  that  a  single  passage 
across  the  line  from  one  edge  to  the  other  makes  a  difference  of  2?ri  in  the 
value,  and  consequently  any  number  of  passages  across  will  give  rise  to  the 
multiplicity  in  value  of  the  integral. 

Such  a  line  is  called  a  section*  by  Hermite,  after  Riemann  who,  in  a 
different  manner,  introduces  these  lines  of  singularity  into  his  method  of 
representing  the  variable  on  surfaces  "f*. 

When  we  take  the  general  integral  of  a  uniform  function  of  Z  and  make 
the  substitution  Z  =  zt,  the  integral  that  arises  for  consideration  is  of  the  form 


We  shall  suppose  that  the  path  of  variation  of  t  is  the  axis  of  real  quantities  : 
and  the  subject  of  integration  will  be  taken  to  be  a  general  function  of  t  and 
z,  without  special  regard  to  its  derivation  from  a  uniform  function  of  Z. 

*  Coupure;  see  Crelle,  t.  xci,  p.  62.  t  See  Chapter  XV. 


103.]  SECTIONS  187 

It  is  easy,  after  the  special  example,  to  see  that  ^  is  a  continuous  function 
of  z  in  any  space  that  does  not  include  a  ^-point  which,  for  values  of  t  included 
within  the  range  of  integration,  would  satisfy  the  equation. 

G  (t,  z)  =  0. 

But  in  the  vicinity  of  a  ^-point,  say  £,  corresponding  to  the  value  t  =  6  in 
the  range  of  integration,  there  will  be  discontinuity  in  the  subject  of 
integration  and  also,  as  will  now  be  proved,  in  the  value  of  the  integral. 

Let  Z  be  the  point  £  and  draw  the  curve  through  Z  corresponding  to 
t  =  real  constant ;  let  Nt  be  a  point  on  the  positive  side  and  N2 
a  point  on  the  negative  side  of  this  curve  positively  described, 
both  points  being  on  the  normal  at  Z ;  and  let 
supposed  small.     Then  for  N!  we  have 

X-L  =  g  —  e  sin  y,         yl  =  ^-\-e  cos  y , 

Fig.  24. 
so  that  z1  =  £+16' (cosy  +  isiny), 

where  ty  is  the  inclination  of  the  tangent  to  the  axis  of  real  quantities.  But, 
if  da-  be  an  arc  of  the  curve  at  Z, 

da  ,  •   •     i  \     d%      •  dt]      d£ 

for  variations  along  the  tangent  at  Z,  that  is, 

i 

da-  .    .  3 

-j--  (cos  y  +  i  sin  y )  =  —  - 


Thus,  since  -j-  may  be  taken  as  finite  on  the  supposition  that  Z  is  an 
ordinary  point  of  the  curve,  we  have 


where  e  =  e'  -y-  ,        P  =  - 

Similarly  z.2  =  £  +  ie  -^r. 

Hence  <1>  (^)  =  I     --i-i— *£  ^ 

w/  n_^J_w/  m  _ 

1*. 


188  HERMITE'S  [103. 

with  a  similar  expression  for  <&  (z2)  ;  and  therefore 

F(t,  0  j-  [G  (t,  ®}^-G  (t,  §) 

' 


The  subject  of  integration  is  infinitesimal,  except  in  the  immediate  vicinity 
of  t  =  6  ;  and  there 


powers  of  small  quantities  other  than  those  retained  being  negligible.  Let 
the  limiting  values  of  t,  that  need  be  retained,  be  denoted  by  d  +  v  and 
d  —  p',  then,  after  reduction,  we  have 

edt 


F(e, 


in  the  limit  when  e  is  made  infinitesimal. 

Hence  a  line  of  discontinuity  of  the  subject  of  integration  is  a  section 
for  the  integral ;  and  the  preceding  expression  is  the  magnitude,  by 
numerical  multiples  of  which  the  values  of  the  integral  differ*. 

Ex.  1.     Consider  the  integral 

dZ 


/ 


zdt 
h 

We  have  S  ^  *'     =^  =  ^g  =  ^. 


so  that  TT  is  the  period  for  the  above  integral. 
Ex.  2.     Shew  that  the  sections  for  the  integral 


ta  sin  z          , 
2     ' 


*  The  memoir  and  the  Cmirs  d' 'Analyse  of  Hermite  should  be  consulted  for  further  develop 
ments;  and,  in  reference  to  the  integral  treated  above,  Jordan,  Cours  d' Analyse,  t.  iii,  pp. 
610 — 614,  may  be  consulted  with  advantage.  See  also,  generally,  for  functions  defined  by 
definite  integrals,  Goursat,  Acta  Math.,  t.  ii,  (1883),  pp.  1—70,  and  ib.,  t.  v,  (1884),  pp.  97— 
120;  and  Pochhammer,  Math.  Arm.,  t.  xxxv,  (1890),  pp.  470—494,  495—526.  Goursat  also 
discusses  double  integrals. 


103.]  SECTIONS  189 


where  a  is  positive  and  less  than  1,  are  the  straight  lines  x  =  (2k  +  l)  TT,  where  k  assumes  all 
integral  values  ;  and  that  the  period  of  the  integral  at  any  section  at  a  distance  77  from  the 
axis  of  real  quantities  is  2?r  cosh  (arj).  (Hermite.) 

Ex.  3.     Shew  that  the  integral 


o 

where  the  real  parts  of  /3  and  y  —  /3  are  positive,  has  the  part  of  the  axis  of  real  quantities 
between  1  and  +00  for  a  section. 

Shew  also  that  the  integral 

i 

rht }—  (z  P~I  (~i  -  vy~'3~1n—    }~a  d 

J  0 

where  the  real  parts  of  /3  and  1  -  a  are  positive,  has  the  part  of  the  axis  of  real  quantities 
between  0  and  1  for  a  section  :  but  that,  in  order  to  render  <£  (z)  a  uniform  function  of  z, 
it  is  necessary  to  prevent  the  variable  from  crossing,  not  merely  the  section,  but  also  the 
part  of  the  axis  of  real  quantities  between  1  and  +  <x> .  (Goursat.) 

(The  latter  line  is  called  a  section  of  the  second  kind.) 

Ex.  4.     Discuss  generally  the  effect  of  changing  the  path  of  t  on  a  section  of  the 

integral ;  and,  in  particular,  obtain  the  section  for    I      —  „  when,  after  the  substitution 

jo  1  +  « 

Z=zt,  the  path  of  t  is  made  a  semi-circle  on  the  line  joining  0  and  1  as  diameter. 

Note.  It  is  manifestly  impossible  to  discuss  all  the  important  bearings  of  theorems  and 
principles,  which  arise  from  time  to  time  in  our  subject ;  we  can  do  no  more  than  mention 
the  subject  of  those  definite  integrals  involving  complex  variables,  which  first  occur  as 
solutions  of  the  better-known  linear  differential  equations  of  the  second  order. 

Thus  for  the  definite  integral  connected  with  the  hypergeometric  series,  memoirs  by 
Jacobi*  and  Goursat  t  should  be  consulted  ;  for  the  definite  integral  connected  with 
Bessel's  functions,  memoirs  by  HankelJ  and  Weber  §  should  be  consulted  ;  and  Heine's 
J/andbuch  der  Kugelfunctionen  for  the  definite  integrals  connected  with  Legendre's 
functions. 

104.     We  shall  now  consider  integrals  of  multiform  functions. 

Ex.  1.  To  find  the  integral  of  a  multiform  function  round  one  loop  ;  and  round  a 
number  of  loops. 

Let  the  function  be 

i 

w={(z-al}(z-a.z}...(z-  an)}»» , 

where  m  may  be  a  negative  or  positive  integer,  and  the  quantities  a  are  unequal  to  one 
another  ;  and  let  the  loop  be  from  the  origin  round  the  point  ax.  Then,  if  /  be  the  value 
of  the  integral  with  an  assigned  initial  branch  w,  we  have 


/a,  f  CO 

wdz-\-  I    wdz  +  I     awdz, 
0  J  c  J  a. 


where  a  is  e  m  and  the  middle  integral  is  taken  round  the  circle  at  a^  of  infinitesimal  radius. 

*  Crelle,  t.  Ivi,  (1859),  pp.  149 — 165 ;  the  memoir  was  not  published  until  after  his  death, 
t  Sur  Vequation  differentielle   lineaire  qui  admet  pour  integrate  la  serie  hypergrometrique, 
(These,  Gauthier-Villars,  Paris,  1881). 

I  Math.  Ann.,  t.  i,  (1869),  pp.  467—501. 

§  Math.  Ann.,  t.  xxxvii,  (1890),  pp.  404—416. 


190 


EXAMPLES 


[104. 


But,  since  the  limit  of  (z-ajw  when  z  =  a1  is  zero,  the  middle  integral  vanishes  by  §  101  ; 
and  therefore 


/"«i 
«,  =  (! -a)  I     web, 

Jo 


where  the  integral  may,  if  convenient,  be  considered  as  taken  along  the  straight  line  from 
0  to  al . 


(2) 


(3) 


Fig.  25. 


Next,  consider  a  circuit  for  an  integral  of  w  which  (fig.  25)  encloses  two  branch-points, 
say  «!  and  «2,  but  no  others  ;  the  circuit  in  (1)  can  be  deformed  into  that  in  (2)  or  into 
that  in  (3)  as  well  as  into  other  forms.  Hence  the  integral  round  all  the  three  circuits 
must  be  the  same.  Beginning  with  the  same  branch  as  in  the  first  case,  we  have 


(1 


/«! 
wdz, 
o 


as  the  integral  after  the  first  loop  in  (2).  And  the  branch  with  which  the  second  loop 
begins  is  aw,  so  that  the  integral  described  as  in  the  second  loop  is 

/«2 
awdz; 
0 

and  therefore,  for  the  circuit  as  in  (2),  the  integral  is 

Cat  [ay 

1=  (1  -  a)  I      wdz  +  a  (1  -  a)  /      wdz. 
Jo  Jo 

Proceeding  similarly  with  the  integral  for  the  circuit  in  (3),  we  find  that  its  expression  is 

/a2  /"<*! 

wdz  +  a  (I -a)  I      wdz, 
0  J  0 

and  these  two  values  must  be  equal. 

But  the  integrals  denoted  by  the  same  symbols  are  not  the  same  in  the  two  cases  ;  the 

function  I   *  wdz  is  different  in  the  second  value  of  J  from  that  in  the  first,  for  the  deforma- 

Jo 

tion  of  path  necessary  to  change  from  the  one  to  the  other  passes  over  the  branch-point  az. 
In  fact,  the  equality  of  the  two  values  of  /  really  determines  the  value  of  the  integral  for 
the  loop  Oal  in  (3). 

And,  in  general,  equations  thus  obtained  by  varied  deformations  do  not  give  relations 
among  loop-integrals  but  define  the  values  of  those  loop-integrals  for  the  deformed  paths. 

We  therefore  take  that  deformation  of  the  circuit  into  loops  which  gives  the  simplest 
path.  Usually  the  path  is  changed  into  a  group  of  loops  round  the  branch-points  as  they 
occur,  taken  in  order  in  a  trigonometrically  positive  direction. 

The  value  of  the  integral  round  a  circuit,  equivalent  to  any  number  of  loops,  is  obvious. 

Ex.  2.  To  find  the  value  of  $wdz,  taken  round  a  simple  curve  which  includes  all  the 
branch-points  of  w  and  all  the  infinities. 


104.]  OF   PERIODICITY   OF   INTEGRALS  191 

If  z  =  oo  be  a  branch-point  or  an  infinity,  then  all  the  branch-points  and  all  the 
infinities  of  w  lie  on  what  is  usually  regarded  as  the  exterior  of  the  curve,  or  the  curve 
may  in  one  sense  be  said  to  exclude  all  these  points.  The  integral  round  the  curve  is  then 
the  integral  of  a  function  round  a  curve,  such  that  over  the  area  included  by  it  the 
function  is  uniform,  finite  and  continuous  ;  hence  the  integral  is  zero. 

If  0  =  00  be  neither  a  branch-point  nor  an  infinity,  the  curve  can  be  deformed  until  it  is 
a  circle,  centre  the  origin  and  of  very  great  radius.  If  then  the  limit  of  zw,  when  \z  is 
infinitely  great,  be  zero,  the  value  of  the  integral  again  is  zero,  by  II.,  §  24. 

Another  method  of  considering  the  integral,  is  to  use  Neumann's  sphere  for  the 
representation  of  the  variable.  Any  simple  closed  curve  divides  the  area  of  the  sphere 
into  two  parts  ;  when  the  curve  is  defined  as  above,  one  of  those  parts  is  such  that  the 
function  is  uniform,  finite  and  continuous  throughout  and  therefore  its  integral  round  the 
curve,  regarded  as  the  boundary  of  that  part,  is  zero.  (See  Corollary  III.,  §  90.) 

Ex.  3.  To  find  the  general  value  of  J(l-22)~*cfe.  The  function  to  be  integrated  is 
two-valued:  the  two  values  interchange  round  each  of  the  branch-points  ±1,  which  are 
the  only  branch-points  of  the  function. 

Let  /  be  the  value  of  the  integral  for  a  loop  from  the  origin  round  +1,  beginning  with 
the  branch  which  has  the  value  +1  at  the  origin  ;  and  let  /'  be  the  corresponding  value 
for  the  loop  from  the  origin  round  -  1,  beginning  with  the  same  branch.  Then,  by  Ex.  1, 

/=  2  P  (1  -  z*T*dz,         /'  =  2  f"1  (1  -  z2)"*  dz 

=  -/, 

the  last  equality  being  easily  obtained  by  changing  variables. 

Now  consider  the  integral  when  taken  round  a  circle,  centre  the  origin  and  of  indefinitely 
great  radius  R  ;  then  by  §  24,  II.,  if  the  limit  of  zw  for  z=  QO  be  k,  the  value  of  \wdz  round 

this  circle  is  2iri&.     In  the  present  case  w  =  (l-  22)~^  so  that  the  limit  of  zw  is  +  ^  ;  hence 

J(l-22r^2  =  27T, 

the  integral  being  taken  round  the  circle.    But  since  a  description  of  the  circle  restores  the 

initial  value,  it  can  be  deformed  into  the  two  loops  from  0  O' 

to  A  and  from  0  to  A'.     The  value  round  the  first  is  /;  and    ^  r          >       ^ 

the  branch  with  which  the  second  begins  to  be  described  has 

the  value  —  1  at  the  origin,  so  that  the  consequent  value  round  *1S-  ^"- 

the  second  is  —  /'  ;  hence 

7-/'  =  2»r* 

and  therefore 

verifying  the  ordinary  result  that 


when  the  integral  is  taken  along  a  straight  line. 

To  find  the  general  value  of  u  for  any  path  of  variation  between  0  and  z,  we  proceed  as 
follows.  Let  Q  be  any  circuit  which  restores  the  initial  branch  of  (l-z2)~^.  Then  by 
§  100,  Corollary  II.,  Q  may  be  composed  of 

(i)  a  set  of  double  circuits  round  +  1,  say  m', 
(ii)  a  set  of  double  circuits  round  -  1,  say  m", 
and  (iii)  a  set  of  circuits  round  +  1  and  -  1  ; 

*  It  is  interesting  to  obtain  this  equation  when  O'  is  taken  as  the  initial  point,  instead  of  0. 


192  EXAMPLES   OF   PERIODICITY  [104. 

and  these  may  come  in  any  order  and  each  may  be  described  in  either  direction.  Now  for 
a  double  circuit  positively  described,  the  value  of  the  integral  for  the  first  description  is  / 
and  for  the  second  description,  which  begins  with  the  branch  —(1  —  z2)~^,  it  is  —  /;  hence 
for  the  double  circuit  it  is  zero  when  positively  described,  and  therefore  it  is  zero  also  when 
negatively  described.  Hence  each  of  the  TO'  double  circuits  yields  zero  as  its  nett  contribu 
tion  to  the  integral. 

Similarly,  each  of  the  m"  double  circuits  round  -  1  yields  zero  as  its  nett  contribution 
to  the  integral. 

For  a  circuit  round  +  1  and  -  1  described  positively,  the  value  of  the  integral  has  just 
been  proved  to  be  /-/',  and  therefore  when  described  negatively  it  is  /'-/.  Hence  if 
there  be  n^  positive  descriptions  and  n2  negative  descriptions,  the  nett  contribution  of  all 
these  circuits  to  the  value  of  the  integral  is  (n±  —  n^)  (I  -  1'),  that  is,  2nir  where  n  is  an 
inteer. 


Hence  the  complete  value  for  the  circuit  Q  i 

Now  any  path  from  0  to  z  can  be  resolved  into  a  circuit  Q,  which  restores  the  initial 
branch  of  (1  —  22)~  ,  chosen  to  have  the  value 
+  1  at  the  origin,  and  either  (i)  a  straight 
line  Oz  ; 

or  (ii)  the  path  OACz,  viz.,  a  loop  round 
+  1  and  the  line  Oz  ; 

or  (iii)  the  path  OA'Cz,  viz.,  a  loop  round 
-  1  and  the  line  Oz. 

Let  u  denote  the  value  for  the  line  Oz,  so  that 

u=  f*  (!-#)-*  dk. 

J  o 

Hence,  for  case  (i),  the  general  value  of  the  integral  is 

2W7T  +  U. 

For  the  path  OA  Cz,  the  value  is  7  for  the  loop  OAC,  and  is  (  —  u)  for  the  line  Cz,  the 
negative  sign  occurring  because,  after  the  loop,  the  branch  of  the  function  for  integra 
tion  along  the  line  is  —(1  —  22)~5  ;  this  value  is  I—u,  that  is,  it  is  TT  —  U.  Hence,  for  case 
(ii),  the  value  of  the  integral  is 

—  U. 


For  the  path  OA'Cz,  the  value  is  similarly  found  to  be  -  TT  -  u  ;  and  therefore,  for  case  (iii), 
the  value  of  the  integral  is 

2?wr  —  ir-u. 

If  /(z)  denote  the  general  value  of  the  integral,  we  have  either 


Or  /(Z)  =  (2TO+1)7T-W, 

where  n  and  m  are  any  integers,  so  that/  (z)  is  a  function  with  two  infinite  series  of  values. 
Lastly,  if  z  =  $($)  be  the  inverse  oif(z}  =  6,  then  the  relation  between  u  and  z  given  by 


can  be  represented  in  the  form 
and 


104.]  OF   INTEGRALS  193 

both  equations  being  necessary  for  the  full  representation.  Evidently  z  is  a  simply -periodic 
function  of  u,  the  period  being  2?r ;  and  from  the  definition  it  is  easily  seen  to  be  an  odd 
function. 

Let  y  =  (\  -z2)—x  (u\  so  that  y  is  an  even  function  of  u  ;  from  the  consideration  of  the 
various  paths  from  0  to  2,  it  is  easy  to  prove  that 


Ex.  4.  To  find  the  general  value  of  f{(l-j*)(l-IM)}~*dk  It  will  be  convenient 
(following  Jordan  *)  to  regard  this  integral  as  a  special  case  of 

Z=  \{(z  -a)(z-  b)  (z  -c}(z-  d)}~*  dz  =  \wdz. 

The  two-valued  function  to  be  integrated  has  a,  6,  c,  d  (but  not  oo )  as  the  complete 
system  of  branch-points  ;  and  the  two  values  interchange  at  each  of  them.  We  proceed  as 
in  the  last  example,  omitting  mere  re-statements  of  reasons  there  given  that  are  applicable 
also  in  the  present  example. 

Any  circuit  Q,  which  restores  an  initial  branch  of  w,  can  be  made  up  of 
(i)  sets  of  double  circuits  round  each  of  the  branch-points, 
and  (ii)  sets  of  circuits  round  any  two  of  the  branch -points. 
The  value  of  \wdz  for  a  loop  from  the  origin  to  a  branch-point  k  (where  k  =  a,b,  c,  or  d)  is 

2  I    wdz  ; 

J  o 
and  this  may  be  denoted  by  K,  where  K=A,  B,  C,  or  D. 

The  value  of  the  integral  for  a  double  circuit  round  a  branch-point  is  zero.  Hence  the 
amount  contributed  to  the  value  of  the  integral  by  all  the  sets  in  (i)  as  this  part  of 
Q  is  zero. 

The  value  of  the  integral  for  a  circuit  round  a  and  b  taken  positively  is  A  -  B  ;  for  one 
round  b  and  c  is  B-  C ;  for  one  round  c  and  d  is  C-D;  for  one  round  a  and  c  is  A  -  C, 
which  is  the  sum  of  A  -  B  and  B-C;  and  similarly  for  circuits  round  a  and  d  and  round 
b  and  d.  There  are  therefore  three  distinct  values,  say  A-B,  B-C,  C-D,  the  values 
for  circuits  round  a  and  b,  b  and  c,  c  and  d  respectively  ;  the  values  for  circuits  round  any 
other  pair  can  be  expressed  linearly  in  terms  of  these  values.  Suppose  then  that  the  part 
of  Q  represented  by  (ii),  when  thus  resolved,  is  the  nett  equivalent  of  the  description  of  m' 
circuits  round  a  and  b,  of  n'  circuits  round  b  and  c,  and  of  I'  circuits  round  c  and  d.  Then 
the  value  of  the  integral  contributed  by  this  part  of  Q  is 


•  which  is  therefore  the  whole  value  of  the  integral  for  Q. 

But  the  values  of  A,  £,  C,  D  are  not  independent  f.  Let  a  circle  with  centre  the  origin 
and  very  great  radius  be  drawn  ;  then  since  the  limit  of  zw  for  |s|  =  oo  is  zero  and  since 
2=  cc  is  not  a  branch-point,  the  value  of  \wdz  round  this  circle  is  zero  (Ex.  2).  The  circle 
can  be  deformed  into  four  loops  round  a,  b,  c,  d  respectively  in  order  ;  and  therefore  the 
value  of  the  integral  is  A  -  B  +  C-  D,  that  is, 


Hence  the  value  of  the  integral  for  the  circuit  fl  is 


where  m  and  n  denote  m'  -  1'  and  n'  -  1'  respectively. 
*  Cours  d'  Analyse,  t.  ii,  p.  343. 

t  For  a  purely  analytical  proof  of  the  following  relation,  see  Greenhill's  Elliptic  Functions 
Chapter  II. 

F-  13 


194  PERIODICITY  [104. 

Now  any  path  from  the  origin  to  z  can  be  resolved  into  Q,  together  with  either 
(i)  a  straight  line  from  0  to  z, 

or    (ii)  a  loop  round  a  and  then  a  straight  line  to  z. 

It  might  appear  that  another  resolution  would  be  given  by  a  combination  of  Q  with,  say,  a 
loop  round  b  and  then  a  straight  line  to  z  ;  but  it  is  resoluble  into  the  second  of  the  above 
combinations.  For  at  C,  after  the  description  of  the  loop  B  ,  introduce  a  double  description 
of  the  loop  A,  which  adds  nothing  to  the  value  of  the  integral  and  does  not  in  the  end 
affect  the  branch  of  w  at  C  ;  then  the  new  path  can  be  regarded  as  made  up  of  (a)  the 
circuit  constituted  by  the  loop  round  b  and  the  first  loop  round  a,  (/3)  the  second  loop  round 
a,  which  begins  with  the  initial  branch  of  w,  followed  by  a  straight  path  to  z.  Of  these 
(a)  can  be  absorbed  into  G,  and  (/3)  is  the  same  as  (ii)  ;  hence  the  path  is  not  essentially 
new.  Similarly  for  the  other  points. 

Let  u  denote  the  value  of  the  integral  with  a  straight  path  from  0  to  z;  then  the 
whole  value  of  the  integral  for  the  combination  of  Q  with  (i)  is  of  the  form 


For  the  combination  of  O  with  (ii),  the  value  of  the  integral  for  the  part  (ii)  of 
the  path  is  J,  for  the  loop  round  a,  +(-«),  for  the  straight  path  which,  owing  to  the 
description  of  the  loop  round  a,  begins  with  -  w  ;  hence  the  whole  value  of  the  integral  is 
of  the  form 


Hence,  if  /  (z)  denote  the  general  value  of  the  integral,  it  has  two  systems  of  values,  each 
containing  a  doubly  -infinite  number  of  terms;  and,  if  z  =  <j>(u)  denote  the  inverse  of 
u  =  f  (z\  we  have 


=  0  {m  (A-B}  +  n(B-C)+A  -  u}, 

where  m  and  n  are  any  integers.     Evidently  z  is  a  doubly-periodic  function  of  u,  with 
periods  A-B  and  B-C. 

Ex.  5.     The  case  of  the  foregoing  integral  which  most  frequently  occurs  is  the  elliptic 
integral  in  the  form  used  by  Legendre  and  Jacobi,  viz.  : 

u  =  J{(1  -  z2)  (1  -  kW)}-*dz  =  \wdz, 
where  k  is  real.      The  branch-points  of   the  function  to  be  integrated  are  1,    -1,  ^ 

and  -L  and  the  values  of  the  integral  for  the  corresponding  loops  from  the  origin  are 

A/ 

A 
2  I    wdz, 

J  o 

r-i  ri 

2  I      wdz—  -2  I    wdz, 
Jo  /• 

I    wdz, 
'' 


and 

Now  the  values  for  the  loops  are  connected  by  the  equation 


*  The  value  for  a  loop  round  b  and  then  a  straight  line  to  z,  just  considered,  is  B  -  u 

=  -(A-B)  + 
being  the  value  in  the  text  with  m  changed  to  m  -  1. 


104.] 


OF   ELLIPTIC   INTEGRALS 


195 


and  so  it  will  be  convenient  that,  as  all  the  points  lie  on  the  axis  of  real  variables,  we 
arrange  the  order  of  the  loops  so  that  this  relation  is  identically  satisfied.  Otherwise, 
the  relation  will,  after  Ex.  1,  be  a  definition  of  the  paths  of  integration  chosen  for  the 
loops. 

Among  the  methods  of  arrangement,  which  secure  the  identical  satisfaction  of  the 


Fig.   28. 


relation,  the  two  in  the  figure*  are  the  simplest,  the  curved  lines  being  taken  straight  in 
the  limit  ;  for,  by  the  first  arrangement  when  k  <  1,  we  have 


and,  by  the  second  when  £  >  1,  we  have 


both  of  which  are  identically  satisfied.     We  may  therefore  take  either  of  them ;  let  the 
former  be  adopted. 

The  periods  are  A-B,  B-C,  (and  C-D,  which  is  equal  to  B-A\  and  any  linear 
combination  of  these  is  a  period:  we  shall  take  A  -  B,  and  B-D.  The  latter,  B-D, 
is  equal  to 

n  r-i 

2  /    wdz -2  I      wdz, 

Jo  Jo 

which,  being  denoted  by  4/f,  gives 

4J5T=4  / 

JO{(1-22)(1_£222)}4 

as  one  period.     The  former,  A-B,  is  equal  to 

2  I    wdz -2  I   wdz, 
Jo  Jo 

i 

/    wdz; 

/k 

1|(1- 


which  is  2 

this,  being  denoted  by  2iK',  gives 


dz 


dz' 


where  £'2  +  £2=l  and  the  relation  between  the  variables  of  the  integrals  is 

i 

Hence  the  periods  of  the  integral  are  4K  and  ZiK'.     Moreover,  A  is  2  I"  wdz,  which  i 

i  J» 

2  /    wdz  +  2  I    wdz  = 
Jo  J  i 

Hence  the  general  value  of  f*  {(I  -  z*)  (I  - 


*  Jordan,  Cours  d'  'Analyse,  t.  ii,  p.  356. 


13—2 


196  PERIODICITY  [104. 


or 


that  is,  2K-u  +  4mK+2niK', 

where  u  is  the  integral  taken  from  0  to  z  along  an  assigned  path,  often  taken  to  be 
a  straight  line  ;  so  that  there  are  two  systems  of  values  for  the  integral,  each  containing 
a  doubly  -infinite  number  of  terms. 

If  z  be  denoted  by  $  (u)  —  evidently,  from  the  integral  definition,  an  odd  function 
of  u  —  ,  then 


so  that  z  is  a  doubly-periodic  function  of  u,  the  periods  being  4A  and  2iK'. 

Now  consider  the  function  ^  =  (1  -zrf.     A  2-path  round  T  does  not  affect  ^  by  way  of 
change,  provided  the  curve  does  not  include  the  point  1  ;  hence,  if  zt  =  x  (u),  we  have 


But  a  z-  path  round  the  point  1  does  change  %  into  —z1;  so  that 

X  («)--*  («+**} 

Hence  x  (u\  which  is  an  even  function,  has  two  periods,  viz.,  4AT  and  2A'  +  2i'A",  whence 

x(u)  =  x(u  +  4mK+  2nK+  2niK'). 
Similarly,  taking  z2  =  (l  -Fs2)*  =  -f  (u),  it  is  easy  to  see  that 


so  that  ^  (u),  which  is  an  even  function,  has  two  periods,  viz.,  2  A'  and  4iK'  ;  whence 


=       u 


The  functions  <£  (u),  x  (u\  ^  (M)  are  of  course  sn  w>  cn  '"">  dn  M  respectively. 
Ex.  6.     To  find  the  general  value  of  the  integral 


The  function   to  be  integrated  has  e^  e2,  e3,  and  co    for  its  branch-points;   and  for 
paths  round  each  of  them  the  two  branches  interchange. 

A   circuit  G  which  restores  the  initial  branch  of  the  function  to  be  integrated  can 
be  resolved  into  : — 

(i)     Sets  of  double  circuits  round  each  of  the  branch-points  alone :   as  before,  the 

value  of  the  integral  for  each  of  these  double  circuits  is  zero. 
(ii)    Sets  of  circuits,  each  enclosing  two  of  the  branch-points :    it  is  convenient  to 
retain   circuits  including  oo   and  en  oo   and  e.2,  oc   and   e3,  the   other   three 
combinations  being  reducible  to  these. 
The  values  of  the  integral  for  these  three  retained  are  respectively 

E!  =  2  f  (4  (z  -  ej  (z  -  e2}  (z  -  e^dz  =  2«1 , 
J  «i 

Ez=2  I    {4(2-e1)(2-e2)(s-e3)}~ick=2a>2, 
J  62 

3         J  ea 

*  The  choice  of  o>  for  the  upper  limit  is  made  on  a  ground  which  will  subsequently  be 
considered,  viz.,  that,  when  the  integral  is  zero,  z  is  infinite. 


104.]  OF   ELLIPTIC   INTEGRALS  197 

and  therefore  the  value  of  the  integral  for  the  circuit  O  is  of  the  form 


But  E^  K2,  E3  are  not  linearly  independent.     The  integral  of  the  function  round  any 

curve   in  the   finite   part   of  the   plane,    which   does   not 

include  el5  e<2  or  e3  within  its  boundary,  is  zero,  by  Ex.  2; 

and  this  curve  can  be  deformed  to  the  shape  in  the  figure, 

until   it   becomes   infinitely  large,    without   changing  the 

value  of  the  integral. 

Since  the  limit  of  zw  for  \z\  =  00  is  zero,  the  value  of 
the  integral  from  oo  '  to  oo  is  zero,  by  §  24,  II.  ;  and  if  the 
description  begin  with  a  branch  w,  the  branch  at  oo  is  -w. 
The  rest  of  the  integral  consists  of  the  sum  of  the  values  Fig.  29. 

round  the  loops,  which  is 


because  a  path  round  a  loop  changes  the  branch  of  w  and  the  last  branch  after  describing  the 
loop  round  e3  is  +w  at  GO',  the  proper  value  (§  90,  in).  Hence,  as  the  whole  integral 
is  zero,  we  have 


or  say  E2  = 

Thus  the  value  of  the  integral  for  any  circuit  Q,  which  restores  the  initial  branch  of  w,  can 
be  expressed  in  any  of  the  equivalent  forms  mE^  n  E3,  m'E^n'E^  m"E2  +  ri'Ez,  where 
the  m's  and  ris,  are  integers. 

Now  any  path  from  co  to  z  can  be  resolved  into  a  circuit  fl,  which  restores  at  oo  the 
initial  branch  of  w,  combined  with  either 

(i)    a  straight  path  from  oo  to  2, 

or        (ii)    a  loop  between  oo  and  e1}  together  with  a  straight  path  from  oo  to  z. 
'  (The  apparently  distinct  alternatives,  of  a  loop  between  oo  and  e2  together  with  a  straight 
:  path  from  oo  to  z  and  of  a  similar  path  round  ea,  are  inclusible  in  the  second  alternative 
above  ;  the  reasons  are  similar  to  those  in  Ex.  5.) 

fx 

If  u  denote  j  ^  {^(z-ej  (z-e2)  (z-e3}}~*dz  when  the  integral  is  taken  in  a  straight 

,  line,  then  the  value  of  the  integral  for  part  (i)  of  a  path   is  u;   and  the  value  of  the 
1  integral  for  part  (ii)  of  a  path  is  El  -  u,  the  initial  branch  in  each  case  for  these  parts  being 
.  the  initial  branch  of  w  for  the  whole  path.     Hence  the  most  general  value  of  the  integral 
for  any  path  is 

+  2no>3  +  u, 


or 


the  two  being  evidently  included  in  the  form 

2mo>1  +  2n(,)3±u. 
If,  then,  we  denote  by  z  =  ft>(u)  the  relation  which  is  inverse  to 


we 

In  the  same  way  as  in  the  preceding  example,  it  follows  that 


where  ^  («)  is  -  {4  (z  -  e^  (z  -  e2)  (z  -  e3)}*. 


198  SIMPLE   PERIODICITY  [104 

The  foregoing  simple  examples  are  sufficient  illustrations  of  the  multi 
plicity  of  value  of  an  integral  of  a  uniform  function  or  of  a  multiform 
function,  when  branch-points  or  discontinuities  occur  in  the  part  of  the  plane 
in  which  the  path  of  integration  lies.  They  also  shew  one  of  the  modes  in 
which  singly-periodic  and  doubly-periodic  functions  arise,  the  periodicity 
consisting  in  the  addition  of  arithmetical  multiples  of  constant  quantities 
to  the  argument.  And  it  is  to  be  noted  that,  as  only  a  single  value  of  z 
is  used  in  the  integration,  so  only  a  single  value  of  z  occurs  in  the 
inversion ;  that  is,  the  functions  just  obtained  are  uniform  functions  of  their 
variables.  To  the  properties  of  such  periodic  functions  we  shall  return  in  the 
succeeding  chapters. 

105.  We  proceed  to  the  theory  of  uniform  periodic  functions,  some 
special  examples  of  which  have  just  been  considered ;  and  limitation  will 
be  made  here  to  periodicity  of  the  linear  additive  type,  which  is  only  a  very 
special  form  of  periodicity. 

A  function  f(z)  is  said  to  be  periodic  when  there  is  a  quantity  &>  such 
that  the  equation 

/(*  +  »)=/(*) 

is  an  identity  for  all  values  of  z.  Then/0  +  nw)  =f(z),  where  n  is  any 
integer  positive  or  negative;  and  it  is  assumed  that  &>  is  the  smallest 
quantity  for  which  the  equation  holds,  that  is,  that  no  submultiple  of  &>  will 
satisfy  the  equation.  The  quantity  u>  is  called  a  period  of  the  function. 

A  function  is  said  to  be  simply-periodic  when  there  is  only  a  single 
period :  to  be  doubly-periodic  when  there  are  two  periods ;  and  so  on,  the 
periodicity  being  for  the  present  limited  to  additive  modification  of  the 
argument. 

It  is  convenient  to  have  a  graphical  representation  of  the  periodicity  of  a 
function. 

(i)  For  simply-periodic  functions,  we 
take  a  series  of  points  0,  A1}  A2,..., 
A-i,  ^4_2,...  representing  0,  w,  2o>,  ...  , 
—  <»,  —  2&>, . . . ;  and  through  these  points 
we  draw  a  series  of  parallel  lines,  dividing 
the  plane  into  bands.  Let  P  be  any 
point  z  in  the  band  between  the  lines 
through  0  and  through  A^\  through  P 
draw  a  line  parallel  to  OAl  and  measure 


each  equal  to  OA^  then  all  the  points     / 

P1}  P2,  ... ,  P_i,  P-2, ...  are  represented 

by  z  +  nco  for  positive  and  negative  integral  values  of  n.    But/ (2  +  &»)=/(•*)] 

and  therefore  the  value  of  the  function  at  a  point  Pn  in  any  of  the  bands  is 


105.] 


DOUBLE   PERIODICITY 


199 


the  same  as  the  value  at  P.  Moreover  to  a  point  in  any  of  the  bands  there 
corresponds  a  point  in  any  other  of  the  bands ;  and  therefore,  owing  to  the 
periodic  resumption  of  the  value  at  the  points  corresponding  to  each  point  P, 
it  is  sufficient  to  consider  the  variation  of  the  function  for  points  within  one 
band,  say  the  band  between  the  lines  through  0  and  through  AI.  A  point  P 
within  the  band  is  sometimes  called  irreducible,  the  corresponding  points  P 
in  the  other  bands  reducible. 

If  it  were  convenient,  the  boundary  lines  of  the  bands  could  be  taken 
through  points  other  than  Al}  A2, ... ;  for  example,  through  points  (m  +  |)  &> 
for  positive  and  negative  integral  values  of  ra.  Moreover,  they  need  not  be 
straight  lines.  The  essential  feature  of  the  graphic  representation  is  the 
division  of  the  plane  into  bands. 

(ii)     For  doubly-periodic  functions  a  similar  method  is  adopted.     Let  &> 
and  co'  be  the  two  periods  of  such  a 
function /(#),  so  that 

/<«.+»)»/(*)-/(•+ <0; 

then         f(z  +  nw  +  n'w)  =f(z), 
where  n  and  n'  are  any  integers  positive 
or  negative. 

For  graphic  purposes,  we  take  points 
0,  A-L,  A2,  ...,  A^i,  A_2,  ...  representing 
0,  ft),  2&),  . . . ,  —  to,  —  2(w,  . . . ;  and  we  take 
another  series  0,  B1}  B2, . . . ,  B_1}  B_2,  . . . 
representing  0,  &)',  2&/, . . . ,  —  ft/,  —  2ft/, . . . ; 
through  the  points  A  we  draw  lines 
parallel  to  the  line  of  points  B,  and 
through  the  points  B  we  draw  lines 
parallel  to  the  line  of  points  A.  The  intersection  of  the  lines  through  An 
and  Bn>  is  evidently  the  point  n&>  +  w'&>',  that  is,  the  angular  points  of  the 
parallelograms  into  which  the  plane  is  divided  represent  the  points  nco  +  n'w 
for  the  values  of  n  and  n'. 

Let  P  be  any  point  z  in  the  parallelogram  OAfi-JS^ ;  on  lines  through  P, 
parallel  to  the  sides  of  the  parallelogram,  take  points  Q1}  Q2, ... ,  Q_lt  Q_2, ... 
such  that  PQl  =  QiQ2=  ...  =  ft)  and  points  Rlt  R2, ... ,  R_lt  R_2, ...  such  that 
PRl  =  R^  —  . . .  —  to' ;  and  through  these  new  points  draw  lines  parallel  to 
the  sides  of  the  parallelogram.  Then  the  variables  of  the  points  in  which 
these  lines  intersect  are  all  represented  by  z  +  mw  +  mV  for  positive  and  nega 
tive  integral  values  of  m  and  m' ;  and  the  point  represented  by  z  +  m^  +  m'a)' 
is  situated  in  the  parallelogram,  the  angular  points  of  which  are  mw  4  mot', 
(m  +  1)  &)  +  mw,  mco  +  (mf  +  1)  ft)',  and  (m  -f  1)  &)  +  (m  +  1)  ft/,  exactly  as  P 
is  situated  in  OA^C^.  But 

/  (z  +  m^  +  Wj  V)  =  /  (z\ 


Fig.  31. 


200  RATIO   OF   THE   PERIODS  [105. 

and  therefore  the  value  of  the  function  at  such  a  point  is  the  same  as  the 
value  at  P.  Since  the  parallelograms  are  all  equal  and  similarly  situated. 
to  any  point  in  any  of  them  there  corresponds  a  point  in  OA^G^B^;  and  the 
value  of  the  function  at  the  two  points  is  the  same.  Hence  it  is  sufficient  to 
consider  the  variation  of  the  function  for  points  within  one  parallelogram,  say, 
that  which  has  0,  &>,  o)  +  «',  &>'  for  its  angular  points.  A  point  P  within 
this  parallelogram  is  sometimes  called  irreducible,  the  corresponding  points 
within  the  other  parallelograms  reducible  to  P ;  the  whole  aggregate  of  the 
points  thus  reducible  to  any  one  are  called  homologous  points.  And  the 
parallelogram  to  which  the  reduction  is  made  is  called  the  parallelogram  of 
periods. 

As  in  the  case  of  simply-periodic  functions,  it  may  prove  convenient  to 
choose  the  position  of  the  fundamental  parallelogram  so  that  the  origin  is 
not  on  its  boundary ;  thus  it  might  be  the  parallelogram  the  middle  points  of 
whose  sides  are  +  £&>,  +  ^co'. 

106.  In  the  preceding  representation  it  has  been  assumed  that  the  line 
of  points  A  is  different  in  direction  from  the  line  of  points  B.  If  &>  =  u  +  iv 
and  to'  =  u'+iv',  this  assumption  implies  that  v'/u'  is  unequal  to  v/u,  and 
therefore  that  the  real  part  of  a>'/ia>  does  not  vanish.  The  justification  of 
this  assumption  is  established  by  the  proposition,  due  to  Jacobi  *  : — 

The  ratio  of  the  periods  of  a  uniform  doubly -periodic  function  cannot  be 
real. 

Let/ (2)  be  a  function,  having  CD  and  CD'  as  its  periods.  If  the  ratio  w'/to 
be  real,  it  must  be  either  commensurable  or  incommensurable. 

If  it  be  commensurable,  let  it  be  equal  to  n'/n,  where  n  and  n'  are 
integers,  neither  of  which  is  unity  owing  to  the  definition  of  the  periods  CD 
and  6Dj. 

Let  n'/n  be  developed  as  a  continued  fraction,  and  let  m'fm  be  the  last 
convergent  before  n'jn,  where  m  and  mf  are  integers.  Then 

n'     m  _    1 

n      m      mn' 
that  is,  mn'  -  m'n  =  1, 

,  1  /  .        U>  ,      .  ..         CD 

so  that  mco  ~  mco  =  -(mn~  run  )  =  -  . 

n x  n 

Therefore  f(z)  =f(z  +  m'co  ~  mco'), 

since  m  and  m'  are  integers ;  so  that 

-,  ,        ~(        co\ 

/(*)-/(' -i- s). 

contravening  the  definition  of  CD  as  a  period,  viz.,  that  no  submultiple  of  co  is  a 
period.  Hence  the  ratio  of  the  periods  is  not  a  commensurable  real  quantity. 

*  Ges.  Werke,  t.  ii,  pp.  25,  26. 


106.]  OF   A   UNIFORM   DOUBLY-PERIODIC    FUNCTION  201 

If  it  be  incommensurable,  we  express  oj'/aj  as  a  continued  fraction.  Let 
p/q  and  p'/q'  be  two  consecutive  convergents :  their  values  are  separated  by 
the  value  of  &>'/&>,  so  that  we  may  write 

v~q+     \q'~q)' 

where  1  >  h  >  0. 

Now  pq  <-  p'q  —  1,  so  that 


-  =  P  +  — 
o>       q      qq 


where  e  is  real  and  |e  <  1  ;  hence 


,  e 

qa)  —pa)  =  —,  &>. 


Therefore  f(z)  =f(z  +  qw  —  pa), 

since  p  and  q  are  integers  ;  so  that 


Now  since  &>'/&>  is  incommensurable,  the  continued  fraction  is  unending.  We 
therefore  take  an  advanced  convergent,  so  that  q'  is  very  large.  Then  €-  &>  is 

a  very  small  quantity  and  z  +  -  &>  is  a  point  infinitesimally  near  to  z,  that 

is,  the  function  /  (V),  under  the  present  hypothesis,  resumes  its  value  at  a 
point  infinitesimally  near  to  z.  Passing  along  the  line  joining  these  two 
points  infinitesimally  near  another,  we  should  have  /  (z)  constant  along  a 
line  and  therefore  (§  37)  constant  everywhere  ;  it  would  thus  cease  to  be  a 
varying  function. 

The  ratio  of  the  periods  is  thus  not  an  incommensurable  real  quantity. 

We  therefore  infer  Jacobi's  theorem  that  the  ratio  of  the  periods  cannot 
be  real.  In  general,  the  ratio  is  a  complex  quantity  ;  it  may,  however,  be  a 
pure  imaginary*. 

COROLLARY.  If  a  uniform  function  have  two  periods  wl  and  &>2  such  that 
a  relation 

mlwl  +  ra2G>2  =  0 

exists  for  integral  values  of  m1  and  ?n2,  the  function  is  only  simply-periodic. 
And  such  a  relation  cannot  exist  between  two  periods  of  a  simply-periodic 
function,  if  m^  and  ra2  be  real  and  incommensurable  ;  for  then  the  function 
would  be  constant. 

*  It  was  proved,  in  Ex.  5  and  Ex.  6  of  §  104,  that  certain  uniform  functions  are  doubly-periodic. 
A  direct  proof,  that  the  ratio  of  the  distinct  periods  of  the  functions  there  obtained  is  not  a  real 
quantity,  is  given  by  Falk,  Acta  Math.,  t.  vii,  (1885),  pp.  197—200,  and  by  Pringsheim,  Math. 
Ann.,  t.  xxvii,  (1886),  pp.  151—157. 


202  UNIFORM  [106. 

Similarly,  if  a  uniform  function  have  three  periods  &>1;  a>.2>  o>3,  connected 

by  two  relations 

..  =  0, 


n1o)1  +  n2a)2  +  n3a)3  =  0, 

where  the  coefficients  m  and  n  are  integers,  then  the  function  is  only  simply- 
periodic. 

107.  The  two  following  propositions,  also  due  to  Jacobi*,  are  important 
in  the  theory  of  uniform  periodic  functions  of  a  single  variable  :  — 

If  a  uniform  function  have  three  periods  w^,  «2,  MS  such  that  a  relation 

m^i  +  m.2&>2  +  m3w3  =  0 

is  satisfied  for  integral  values  ofmlt  w2,  m3,  then  the  function  is  only  a  doubly- 
periodic  function. 

What  has  to  be  proved,  in  order  to  establish  this  proposition,  is  that  two 
periods  exist  of  which  wl,  &>2,  &>3  are  integral  multiple  combinations. 

Evidently  we  may  assume  that  m^,  ra2,  m3  have  no  common  factor:  let  / 
be  the  common  factor  (if  any)  of  m.2  and  m3,  which  is  prime  to  m^.  Then 
since 


and  the  right-hand  side  is  an  integral  combination  of  periods,  it  follows  that 
riod. 

is  a  fraction  in  its  lowest  terms.     Change  it  into  a  continued 


-~  &>!  is  a  period. 


fraction  and  let  ^  be  the  last  convergent  before  the  proper  value  ;  then 
2 


1 

so  that  <l~f~P=±^f- 

But  o>!  is  a  period  and  ^ft)!  is  a  period;  therefore  q  —^  Wj  —  pwi  is  a  period, 

or  &>!//  is  a  period,  =  to/  say. 

Let   ra2//=  m2',    m3/f=  m/,    so    that    m1&V  +  m2'&>2  +  ??i3'&)3  =  0.     Change 

fy» 

m.2'/m3  into  a  continued  fraction,  taking  -  to  be  the  last  convergent  before  the 

proper  value,  so  that 

m/      r  _        1 

/  i 


s         sms 

*   Ges.  Werke,  t.  ii,  pp.  27—32. 


107.]  DOUBLY-PERIODIC    FUNCTIONS  203 

Then  r&>2  +  sco.,  being  an  integral  combination  of  periods,  is  a  period.     But 
±  &>2  =  &)2  (sm2r  —  rm3) 

=  —  ra>.2m3  —  s  (m^  +  w3'&>3) 

=  —  m^sw-i'  -  ma'  (r&>2  +  su>3)  ; 
also  +  ft)3  =  &)3  (sm/  —  rm3) 

—  sm2'o)3  +  r  (mjO)/ 


and  o>!  =/&)/. 

Hence  two  periods  &>/  and  r<u2  +  s&>3  exist  of  which  co1}  co2,  &>3  are  integral 
multiple  combinations  ;  and  therefore  all  the  periods  are  equivalent  to  &>/  and 
r&>2  +  so)3,  that  is,  the  function  is  only  doubly-periodic. 

COROLLARY.  If  a  function  have  four  periods  <ul3  &>2,  cos,  &>4  connected  by 
two  relations 

m1o)1  +  m2o)2  +  wi3ft)3  +  ra4&>4  =  0, 

72J60J    +  W20)2    +  W3ft)3    +  W4«04    =  0, 

where  the  coefficients  m  and  w  are  integers,  the  function  is  only  doubly- 
periodic. 

108.  If  a  uniform  function  of  one  variable  have  three  periods  a)l,  w.,,  &>3, 
then  a  relation  of  the  form 

m1o)l  +  w?2to2  +  in3(i)3  =  0 
must  be  satisfied  for  some  integral  values  ofml}  m2,  ms. 

Let  a)r  =  ar  +  i@r,  for  r  =  1,  2,  3  ;  in  consequence  of  §  106,  we  shall  assume 
that  no  one  of  the  ratios  of  twj,  <w2,  w3  in  pairs  is  real,  for,  otherwise,  either 
the  three  periods  reduce  to  two  immediately,  or  the  function  is  a  constant. 
Then,  determining  two  quantities  A,  and  fj,  by  the  equations 


so  that  X  and  //,  are  real  quantities  and  neither  zero  nor  infinity,  we  have 


for  real  values  of  X  and  p. 

Then,  first,  if  either  X  or  fj.  be  commensurable,  the  other  is  also  commen 
surable.     Let  X  =  a/6,  where  a  and  b  are  integers  ;  then 


=  bo)3  —  aa)}, 

so  that  fyu,&>2  is  a  period.  Now,  if  b/j,  be  not  commensurable,  change  it  into  a 
continued  fraction,  and  let  p/q,  p'/q  be  two  consecutive  convcrgents,  so  that, 
as  in  §  106, 

/        P  ,    x 
bfji,=^+  —,, 

q      qq 


204  TRIPLY-PERIODIC    UNIFORM  [108. 

where  1  >  x  >  —  1.     Then  -  &>.,  +  -— ?  is  a  period,  and  so  is  <w2 ;  hence 

q  qq 

'P~  ^x 


IT 

is  a  period,  that  is,  -  <a2  is  a  period.     We  may  take  q  indefinitely  large,  and 

then  the  function  has  an  infinitesimal  quantity  for  a  period,  that  is,  it  would 
be  a  constant  under  the  hypothesis.  Hence  &/*  (and  therefore  /*)  cannot  be 
incommensurable,  if  X  be  commensurable;  and  thus  X  and  //.  are  simul 
taneously  commensurable  or  simultaneously  incommensurable. 

CL  G 

If  X  and  fj,  be  simultaneously  commensurable,  let  X  =  j-  ,  p  =  -^  ,  so  that 

a  c 

&)3  =  r  &>!  +   -jG>2. 

o  a 

and  therefore  6rfto3  =  ac^  +  bca)2, 

a  relation  of  the  kind  required. 

If  X  and  //.  be  simultaneously  incommensurable,  express  A,  as  a  continued 
fraction  ;  then  by  taking  any  convergent  r/s,  we  have 

r  _  x 
*=*' 

/Yt 

where  1  >  x  >  —  1,  so  that  s\  —  r=-: 

s 

by  taking  the  convergent  sufficiently  advanced  the  right-hand  side  can  be 
made  infinitesimal. 

Let  i\  be  the  nearest  integer  to  the  value  of  s/j,,  so  that,  if 


we  have  A  numerically  less  than  ^.     Then 

x 

sat-,  —  ra>1  —  r1w2  =  —  a)1  +  Aw.,, 
s 

fp 

and  the  quantity  -  Wj  can  be  made  so  small  as  to  be   negligible.     Hence 

S 

integers  r,  rlt  s  can  be  chosen  so  as  to  give  a  new  period  &>/(=  A&>2),  such 
that  |  &)/  <  \  &)2  . 

We  now  take  wl,  &>2',  &>3:  they  will  be  connected  by  a  relation  of  the  form 

0>3  =X'(W1  +yLt/G)2/, 

and  X'  and  //  must  be  incommensurable  :  for  otherwise  the  substitution  for 
to/  of  its  value  just  obtained  would  lead  to  a  relation  among  a>l)  &>o,  &>3  that 
would  imply  commensurability  of  X  and  of  p. 

Proceeding  just  as  before,  we  may  similarly  obtain  a  new  period  &>2"  such 
that   <o2"  <  \  !  mz    I  and  so  on  in  succession.     Hence  we  shall  obtain,  after  n 


108.]  FUNCTIONS    DO    NOT   EXIST  205 

such  processes,  a  period  co2(w)  such  that  |&)2(n)|  <  ^  a>*\,  so  that  by  making  n 

z 

sufficiently  large  we  shall  ultimately  obtain  a  period  less  than  any  assigned 
quantity.  Let  such  period  be  to  ;  then 

/(*+«)-/(*), 

and  so  for  points  along  the  co-line  we  have  an  infinite  number  close  together 
at  which  the  function  is  unaltered  in  value.  The  function,  being  uniform, 
must  in  that  case  be  constant. 

It  thus  appears  that,  if  A.  and  /j,  be  simultaneously  incommensurable,  the 
function  is  a  constant.  Hence  the  only  tenable  result  is  that  A.  and  //.  are 
simultaneously  commensurable,  and  then  there  is  a  period-equation  of  the 

form 

m^w^  +  m.2o)2  +  m3o)s  =  0, 

where  m1,  w2,  m3  are  integers. 

The  foregoing  proof  is  substantially  due  to  Jacobi  (I.e.).  The  result  can 
be  obtained  from  geometrical  considerations  by  shewing  that  the  infinite 
number  of  points,  at  which  the  function  resumes  its  value,  along  a  line 
through  z  parallel  to  the  two-line  will,  unless  the  condition  be  satisfied,  reduce 
to  an  infinite  number  of  points  in  the  a)1,  &)2  parallelogram  which  will  form 
either  a  continuous  line  or  a  continuous  area,  in  either  of  which  cases  the 
function  would  be  a  constant.  But,  if  the  condition  be  satisfied,  then  the 
points  along  the  line  through  z  reduce  to  only  a  finite  number  of  points. 

COROLLARY  I.  Uniform  functions  of  a  single  variable  cannot  have  three 
independent  periods ;  in  other  words,  triply -periodic  uniform  functions  of  a 
single  variable  do  not  exist* ;  and,  a  fortiori,  uniform  functions  of  a  single 
variable  with  a  number  of  independent  periods  greater  than  two  do  not  exist. 

But  functions  involving  more  than  one  variable  can  have  more  than  two 
periods,  e.g.,  Abelian  transcendents ;  and  a  function  of  one  variable,  having 
more  than  two  periods,  is  not  uniform. 

COROLLARY  II.  All  the  periods  of  a  uniform  periodic  function  of  a  single 
variable  reduce  either  to  integral  multiples  of  one  period  or  to  linear  combina 
tions  of  integral  multiples  of  two  periods  whose  ratio  is  not  a  real  quantity. 

109.  It  is  desirable  to  have  the  parallelogram,  in  which  a  doubly- 
periodic  function  is  considered,  as  small  as  possible.  If  in  the  parallelogram 
(supposed,  for  convenience,  to  have  the  origin  for  an  angular  point)  there  be 
a  point  a)"  such  that 

/(*  +  »")=/(*) 
for  all  values  of  z,  then  the  parallelogram  can  be  replaced  by  another. 

*  This  theorem  is  also  due  to  Jacobi,  (I.e.,  p.  202,  note). 


206  FUNDAMENTAL    PARALLELOGRAM  [109. 

It  is  evident  that  co"  is  a  period  of  the  function  ;  hence  (§  108)  we  must 

have 

co"  =  Aco  +  /AW'  ; 

and  both  X  and  /JL,  which  are  commensurable  quantities,  are  less  than  unity 
since  the  point  is  within  the  parallelogram.  Moreover,  co  -f  co'  —  <»",  which 
is  equal  to  (1  —  A,)  co  +  (1  —  /"•)&>',  is  another  point  within  the  parallelogram; 
and 

/(*  +  »  +  «'-«")«/(*), 

since  co,  co',  co"  are  periods.     Thus  there  cannot  be  a  single  such  point,  unless 

X  =  \  =  p. 

But  the  number  of  such  points  within  the  parallelogram  must  be  finite  ; 
if  there  were  an  infinite  number,  they  would  form  a  continuous  line  or  a 
continuous  area  where  the  uniform  function  had  an  unvarying  value,  and 
consequently  (§  37)  the  function  would  have  a  constant  value  everywhere. 

To  construct  a  new  parallelogram  when  all  the  points  are  known,  we  first 
choose  the  series  of  points  parallel  to  the  co-line  through  the  origin  0,  and  of 
that  series  we  choose  the  point  nearest  0,  say  Al.  We  similarly  choose  the 
point,  nearest  the  origin,  of  the  series  of  points  parallel  to  the  co-line  and 
nearest  to  it  after  the  series  that  includes  Al}  say  Bl  :  we  take  OA1}  OB1  as 
adjacent  sides  of  the  parallelogram,  and  these  lines  as  the  vectorial  repre 
sentations  of  the  periods.  No  point  lies  within  this  parallelogram  where  the 
function  has  the  same  value  as  at  0  ;  hence  the  angular  points  of  the  original 
parallelograms  coincide  with  angular  points  of  the  new  parallelograms. 

When  a  parallelogram  has  thus  been  obtained,  containing  no  internal 
point  fl  such  that  the  function  can  satisfy  the  equation 


for  all  values  of  z,  it  is  called  a  fundamental,  or  a  primitive,  parallelogram,  : 
and  the  parallelogram  of  reference  in  subsequent  investigations  will  be 
assumed  to  be  of  a  fundamental  character. 

But  a  fundamental  parallelogram  is  not  unique. 

Let  co  and  co'  be  the  periods  for  a  given  fundamental  parallelogram,  so 
that  every  other  period  co"  is  of  the  form  Aco  +  //-co',  where  A,  and  /*  are 
integers.  Take  any  four  integers  a,  b,  c,  d  such  that  ad  —  lc=±l,  as  may 
be  done  in  an  infinite  variety  of  ways  ;  and  adopt  two  new  periods  coj  and  co2, 
such  that 

&>!  =  aco  +  bo)',         co2  =  ceo  +  d(o'. 

Then  the  parallelogram  with  coj  and  co2  for  adjacent  sides  is  fundamental. 
For  we  have 

+  eo  =  do)1  —  ba>2,        +  co'  =  —  ccox  +  aco2, 

and  therefore  any  period  co" 
=  A.CO  +  /uco' 
=  (\d  -  fie)  wl  +  (—  \b  +  fj.a)  eo2,  save  as  to  signs  of  A,  and  /z. 


109.]  OF   PERIODS  207 

The  coefficients  of  o^  and  &)2  are  integers,  that  is,  the  point  <w"  lies  outside 
the  new  parallelogram  of  reference;  there  is  therefore  no  point  in  it  such  that 

/(*  +  *>")=/(*), 
and  hence  the  parallelogram  is  fundamental. 

COROLLARY.  The  aggregate  of  the  angular  points  in  one  division  of  the 
plane  into  fundamental  parallelograms  coincides  with  their  aggregate  in 
any  other  division  into  fundamental  parallelograms  ;  and  all  fundamental 
parallelograms  for  a  given  function  are  of  the  same  area. 

The  method  suggested  above  for  the  construction  of  a  fundamental  parallelogram  is 
geometrical,  and  it  assumes  a  knowledge  of  all  the  points  w"  within  a  given  parallelogram 
for  which  the  equation/  (z  -f  «")=/  (z)  is  satisfied. 

Such  a  point  o>3  within  the  o^,  o>2  parallelogram  is  given  by 

nil          m2 

<Bo=  -  (Bi   -\  --  0>9, 

'     m3    J      m3    2 

where  »&1}  m2,  m3  are  integers.      We  may  assume  that  no  two  of  these  three   integers 
have   a  common   factor;    were  it  otherwise,  say  for  m^  and  wi2,  then,  as  in   §   107,  a 
submultiple  of  o>3  would  be  a  period  —  a  result  which  may  be  considered  as  excluded. 
Evidently  all  the  points  in  the  parallelogram  are  the  reduced  points  homologous  with 
w3,    2o>3,  ......  ,   (m3  —  1)«3;    when   these   are    obtained,  the  geometrical   construction   is 

possible. 

The  following  is  a  simple  and  practicable  analytical  method  for  the  construction. 

Change  w^/rag  and  mz/m3  into  continued  fractions;  and  let  p/q  and  r/s  be  the 
last  convergents  before  the  respective  proper  values,  so  that 

mx     p        e  m2     r       f' 

m3     q     gm3'        m3     s      sm3' 
where  e  and  e'  are  each  of  them  +1.     Let 

m">       n  ,    M  ml        j    ,     ^ 

q  —  =d  +  —  ,      s^  =  $+  —  , 

m3  m3          m3  m3 

where  X  and  p,  are  taken  to  be  less  than  m3,  but  they  do  not  vanish  because  q  and  s  are 
less  than  m3.  Then 

2'eo3-^w1-(9o)2  =  —  (/*a>2  +  f»i),  *a>3-ro>2  -<£<•>!  =  —  (Xa^  +  e'tOjj)  ; 

U  vn  II  io 

the  left-hand  sides  are  periods,  say  Qx  and  O2  respectively,  and  since  /u  +  e  is  not  >m3  and 
X  +  e'  is  not  >m3,  the  points  Q.l  and  Q2  determine  a  parallelogram  smaller  than  the  initial 
parallelogram. 


Thus 
are  equations  defining  new  periods  Qly  Q2.     Moreover 

,  .   X         m-.        p       65  a         m9        r      t'o 

4>-\  --  =  s-^=s*-+      -,  0  +  -f^  =  n  —?  =  «-  +  -L  : 

m3        m3        q      qms  m3     2  ms     *  s      sm3 

so  that,  multiplying  the  right-hand  sides  together  and  likewise  the  left-hand  sides,  we 
at  once  see  that  X/i-ee'  is  divisible  by  ms  if  it  be  not  zero:   let 

X/i  —  ee'  =  wi3A. 

Then,  as  X  and  p  are  less  than  m3,  they  are  greater  than  A;  and  they  are  prime  to  it, 
because  ee'  is   +1. 


208  MULTIPLE  [109. 

Hence  we  have  Aa>j  =  ^Q2-  t'Ql,         Aa>2  =  XQ1-  eiV 

Since  X  and  /u  are  both  greater  than  A,  let 

X  =  X1A  +  X',         /x  =  /i1A  +  //, 
where  X'  and  /x'  are  <A.     Then  X'/*'—  «'  ig  divisible  by  A  if  it  be  not  zero,  say 

X'p  -  ee  =  AA'  ; 
then  X'  and  p.'  are  >A'  and  are  prime  to  it.     And  now 

A  (wj  —  /^iO2)  =  /x'Q2  ~  e'^i  >  A  (W2  "~  ^1^1)  =  ^-'QI  ~  f®2  i 

and  therefore,  if  (a1  —  /^G^Qg,  <B2-X1Q1  =  Q4,  which  are  periods,  we  have 


With  Q3  and  Q4  we  can  construct  a  parallelogram  smaller  than  that  constructed 
with  Qj  and  Q2. 

We  now  have  A'Q1  =  fG3+//G4,         A'Q.j=X'Q3  +  e'fl4, 

that  is,  equations  of  the  same  form  as  before.  We  proceed  thus  in  successive  stages  : 
each  quantity  A  thus  obtained  is  distinctly  less  than  the  preceding  A,  and  so  finally  we 
shall  reach  a  stage  when  the  succeeding  A  would  be  unity,  that  is,  the  solution  of  the  pair 
of  equations  then  leads  to  periods  that  determine  a  fundamental  parallelogram.  It 
is  not  difficult  to  prove  that  a>lt  o>2,  o>3  are  combinations  of  integral  multiples  of  these 
periods. 

If  one  of  the  quantities,  such  as  X'/x'-ee',  be  zero,  then  X'=/x'  =  l,  e  =  e'=  ±1  ;  and  then 
Q3  and  O4  are  identical.  If  e  =  e'  =  +  1,  then  AQ3  =  Q2  -  Qj  ,  and  the  fundamental  parallelo 
gram  is  determined  by 

<V  =  QI  +  -  (Q2  -  %),  G4'  =  Q2  -  1  (Q2  -  Qt)- 

If  f  =  f  =  -1,  then  AQ3  =  Q2+O15  so  that,  as  A  is  not  unity  in  this  case,  the  fundamental 
parallelogram  is  determined  by  Q2  and  Q3. 

Ex.     If  a  function  be  periodic  in  a>1?  a>2,  and  also  in  <o3  where 

29co  =  1  7 


periods  for  a  fundamental  parallelogram  are 

QI  =  Scoj  +  3o)2  -  8w3  ,          Q2'  =  3  eoj  +  2co2  -  5w3  , 
and  the  values  of  a>1}  <»2,  w3  in  terms  of  O/  and  Q2'  are 


G)2  =      2-1,          a>3  =      2      Q. 

Further  discussion  relating  to  the  transformation  of  periods  and  of  fundamental 
parallelograms  will  be  found  in  Briot  and  Bouquet's  The'orie  des  fonctions  elliptujues, 
pp.  234,  235,  268—272. 

110.  It  has  been  proved  that  uniform  periodic  functions  of  a  single 
variable  cannot  have  more  than  two  periods,  independent  in  the  sense  that 
their  ratio  is  not  a  real  quantity.  If  then  a  function  exist,  which  has  two 
periods  with  a  real  incommensurable  ratio  or  has  more  than  two  independent 
periods,  either  it  is  not  uniform  or  it  is  a  function  (whether  uniform  or  multi 
form)  of  more  variables  than  one. 

When  restriction  is  made  to  uniform  functions,  the  only  alternative  is 
that  the  function  should  depend  on  more  than  one  variable. 


110.]  PERIODICITY  209 

In  the  case  when  three  periods  o)l,  &>2,  &>3  (each  of  the  form  a-f  t/3)  were 
assigned,  it  was  proved  that  the  necessary  condition  for  the  existence  of  a 
uniform  function  of  a  single  variable  is  that  finite  integers  mly  m2,  m:i  can 
be  found  such  that 

ra2cr2  +  m3a3  =  0, 

-  w3/33  =  0  ; 

and  that,  if  these  conditions  be  not  satisfied,  then  finite  integers  m1}  m.2,  ms 
can  be  found  such  that  both  Sma  and  2m/8  become  infinitesimally  small. 

This  theorem  is  purely  algebraical,  and  is  only  a  special  case  of  a  more 
general  theorem  as  follows : 

Let  an,  or12,...,  alt  r+1;  a21,  aw,...,  a2>r+1;...;  arl,  «.«,...,  ar,r+i  be  r  sets  of 
real  quantities  such  that  a  relation  of  the  form 

wia*i  +  ^2^2  +  .  •  •  +  nr+l  <xsr+i  =  0 

is  not  satisfied  among  any  one  set.  Then  finite  integers  m^,...,  mr+1  can  be 
determined  such  that  each  of  the  sums 

j  (for  5  =  1,  2,...,r)  is  an  infinitesimally  small  quantity.  And,  a  fortiori,  if 
fewer  than  r  sets,  each  containing  r+1  quantities  be  given,  the  r+1 
integers  can  be  determined  so  as  to  lead  to  the  result  enunciated ;  all  that 

i  is  necessary  for  the  purpose  being  an  arbitrary  assignment  of  sets  of  real 

|  quantities  necessary  to  make  the  number  of  sets  equal  to  r.     But  the  result 

!  is  not  true  if  more  than  r  sets  be  given. 

We  shall  not  give  a  proof  of  this  general  theorem*  ;  it  would  follow  the 
lines  of  the  proof  in  the  limited  case,  as  given  in  §  108.  But  the  theorem 
will  be  used  to  indicate  how  the  value  of  an  integral  with  more  than 
two  periods  is  affected  by  the  periodicity. 

Let  /  be  the  value  of  the  integral  taken  along  some  assigned  path  from 
an  initial  point  ZQ  to  a  final  point  z\  and  let  the  periods  be  &)1}  &>2,...,  &>r, 
(where  r  >  2),  so  that  the  general  value  is 

/  +  fftjcoj  +  m2a).,  +  . . .  +  mrwr, 

where  mlt  m2,...,  mr  are  integers.  Now  if  cos  =  as  +  i/3s,  for  s=l,  2,...,  r, 
when  it  is  divided  into  its  real  and  its  imaginary  parts,  then  finite  integers 
Wi,  n2,...,  nr  can  be  determined  such  that 


and  n-if. 

are  both  infinitesimal ;  and  then 


2  ns 


is  infinitesimal.     But  the  addition 


of  S  nscos  still  gives  a  value  of  the  integral ;  hence  the  value  can  be  modified 

*  A  proof  will  be  found  in  Clebsch  and  Gordan's  Theorie  der  Abel'schen  Functioncn,  §  38. 
F-  14 


210  MULTIPLE   PERIODICITY  [110. 

by  infinitesimal  quantities,  and  the  modification  can  be  repeated  indefinitely. 
The  modifications  of  the  value  correspond  to  modifications  of  the  path  from 
ZQ  to  z ;  and  hence  the  integral,  regarded  as  depending  on  a  single  variable, 
can  be  made,  by  modifications  of  the  path  of  the  variable,  to  assume  any 
value.  The  integral,  in  fact,  has  not  a  definite  value  dependent  solely 
upon  the  final  value  of  the  variable;  to  make  the  value  definite,  the  path 
by  which  the  variable  passes  from  the  lower  to  the  upper  limit  must  be 
specified. 

It  will  subsequently  (§  239)  be  shewn  how  this  limitation  is  avoided  by 
making  the  integral,  regarded  as  a  function,  depend  upon  a  proper  number 
of  independent  variables — the  number  being  greater  than  unity. 

Ex.  1.     If  F0  be  the  value  of   i  — -,  ,  (n  integral),  taken  along  an  assigned  path, 

Jo  (\-znY 
and  if 

P  =  2  I1—- ^-j(#  real), 

then  the  general  value  of  the  integral  is 

I  \        ^  I 

n 

where  q  is  any  integer  and  mp  any  positive  or  negative  integer  such  that  2  mp  =  0. 

P=I 

(Math.  Trip.  Part  II,  1889.) 

Ex.  2.     Prove  that  v=  I   udz,  where 
J  o 

is  an  algebraical  function  satisfying  the  equation 

and  obtain  the  conditions  necessary  and  sufficient  to  ensure  that 

i)  =  fadz 
should  be  an  algebraical  function,  when  u  is  an  algebraical  function  satisfying  an  equation 

(Liouville,  Briot  and  Bouquet.) 


CHAPTER  X. 


SIMPLY-PERIODIC  AND  DOUBLY-PERIODIC  FUNCTIONS. 


111.  ONLY  a  few  of  the  properties  of  simply-periodic  functions  will  be 
given,  partly  because  some  of  them  are  connected  with  Fourier's  series  the 
detailed  discussion  of  which  lies  beyond  our  limits,  and  partly  because,  as 
will  shortly  be  explained,  many  of  them  can  at  once  be  changed  into 
properties  of  uniform  non-periodic  functions  which  have  already  been 
considered. 

When  we  use  the  graphical  method  of  §  105,  it  is  evident  that  we  need 
consider  the  variation  of  the  function  within  only  a  single  band.  Within 
that  band  any  function  must  have  at  least  one  infinity,  for,  if  it  had  not,  it 
would  not  have  an  infinity  anywhere  in  the  plane  and  so  would  be  a  constant ; 
and  it  must  have  at  least  one  zero,  for,  if  it  had  not,  its  reciprocal,  also  a 
simply-periodic  function,  would  not  have  an  infinity  in  the  band.  The 
infinities  may,  of  course,  be  accidental  or  essential :  their  character  is  repro 
duced  at  the  homologous  points  in  all  the  bands. 

For  purposes  of  analytical  representation,  it  is  convenient  to  use  a 
relation 

Ziri 

so  that,  if  the  point  Z  in  its  plane  have  R  and  (*) 
for  polar  coordinates, 


,       „ 

Z  =  =—  ;  log  R  + 

Z7TI 


ft). 


If  we  take  any  point  A  in  the  ^-plane  and  a 
corresponding  point  a  in  the  z-plane,  then,  as  Z 
describes  a  complete  circle  through  A  with  the 
origin  as  centre,  z  moves  along  a  line  aal}  where 
di  is  a  +  a).  A  second  description  of  the  circle 
makes  z  move  from  ax  to  aa,  where  a2  =  ax  +  &>  • 


Fig.  32. 

and  so  on  in  succession. 
14—2 


212  SIMPLE   PERIODICITY  [111. 

For  various  descriptions,  positive  and  negative,  the  point  a  describes  a  line, 
the  inclination  of  which  to  the  axis  of  real  quantities  is  the  argument  of  &>. 

Instead  of  making  Z  describe  a  circle  through  A,  let  us  make  it  describe 
a  part  of  the  straight  line  from  the  origin  through  A,  say  from  A,  where 
OA  =  R,  to  C,  where  00  =  R'.  Then  z  describes  a  line  through  a  perpend 
icular  to  aal}  and  it  moves  to  c  where 


Similarly,  if  any  point  A'  on  the  former  circumference  move  radially  to  a 
point  C  at  a  distance  R  from  the  ^-origin,  the  corresponding  z  point  a' 
moves  through  a  distance  a'c',  parallel  and  equal  to  ac  :  and  all  the  points  c' 
lie  on  a  line  parallel  to  aa^.  Repeated  description  of  a  ^-circumference  with 
the  origin  as  centre  makes  z  describe  the  whole  line  cCjCo. 

If  then  a  function  be  simply-periodic  in  &>,  we  may  conveniently  take 
any  point  a,  and  another  point  a^  =  a  +  w,  through  a  and  a^  draw  straight 
lines  perpendicular  to  aa1}  and  then  consider  the  function  within  this  band. 
The  aggregate  of  points  within  this  band  is  obtained  by  taking 

(i)     all  points  along  a  straight  line,  perpendicular  to  a  boundary  of 

the  band,  as  aa^  ; 
(ii)    the  points  along  all  straight  lines,  which  are  drawn  through  the 

points  of  (i)  parallel  to  a  boundary  of  the  band. 

In  (i),  the  value  of  z  varies  from  0  to  co  in  an  expression  a  +  z,  that  is,  in 
the  ^-plane  for  a  given  value  of  R,  the  angle  ©  varies  from  0  to  2?r. 

In   (ii),  the  value  of  log  R  varies   from   —  oo    to  +00  in  an  expression 
fi\ 

.  log  R  +  =—  w,  that  is,  the  radius  R  must  vary  from  0  to  oo  . 

2?r 


Hence  the  band  in  the  0-plane  and  the  whole  of  the  ^-plane  are  made 
equivalent  to  one  another  by  the  transformation 


Now  let  z0  be  any  special  point  in  the  finite  part  of  the  band  for  a  given 
simply-periodic  function,  and  let  Z0  be  the  corresponding  point  in  the  Z-planej 
Then  for  points  z  in  the  immediate  vicinity  of  z0  and  for  points  Z  which 
are  consequently  in  the  immediate  vicinity  of  Z0,  we  have 


Ziri 


e 
to 


where  |  X    differs  from  unity  only  by  an  infinitesimal  quantity. 


111.]  FOURIER'S  THEOREM  213 

If  then  w,  a  function  of  z,  be  changed  into  W  a  function  of  Z,  the  following 
relations  subsist : — 

When  a  point  ZQ  is  a  zero  of  w,  the  corresponding  point  ZQ  is  a  zero 

of  W. 

When  a  point  z0  is  an  accidental  singularity  of  w,  the  corresponding 
point  Z0  is  an  accidental  singularity  of  W. 

When  a  point  z0  is  an  essential  singularity  of  w,  the  corresponding 
point  Z0  is  an  essential  singularity  of  W. 

When  a  point  z0  is  a  branch- point  of  any  order  for  a  function  w,  the 
corresponding  point  Z0  is  a  branch-point  of  the  same  order  for  W. 

And  the  converses  of  these  relations  also  hold. 

Since  the  character  of  any  finite  critical  point  for  w  is  thus  unchanged  by  the 
transformation,  it  is  often  convenient  to  change  the  variable  to  Z  so  as  to  let 
the  variable  range  over  the  whole  plane,  in  which  case  the  theorems  already 
proved  in  the  preceding  chapters  are  applicable.  But  special  account  must 
be  taken  of  the  point  z  =  oo  . 

112.     We  can  now  apply  Laurent's  theorem  to  deduce  what  is  practically 

Fourier's  series,  as  follows. 

Let  f(z)  be  a  simply-periodic  function  having  w  as  its  period,  and  suppose 
that  in  a  portion  of  the  z-plane  bounded  by  any  two  parallel  lines,  the  inclina 
tion  of  which  to  the  axis  of  real  quantities  is  equal  to  the  argument  of  w,  the 
function  is  uniform  and  has  no  singularities;  then,  at  points  within  that 
portion  of  the  plane,  the  function  can  be  expressed  in  the  form  of  a  converging 

2n-2t 

series  of  positive  and  of  negative  integral  powers  of  e  "°  . 

In  figure  32,  let  aa^a^...  and  cc^...  be  the  two  lines  which  bound  the 
portion  of  the  plane :  the  variations  of  the  function  will  all  take  place  within 
that  part  of  the  portion  of  the  plane  which  lies  within  one  of  the  repre 
sentative  bands,  say  within  the  band  bounded  by  ...ac...  and  . ..a^...:  that  is, 
we  may  consider  the  function  within  the  rectangle  acc^a,  where  it  has  no 
singularities  and  is  uniform. 

Now  the  rectangle  acc^a  in  the  2-plane  corresponds  to  a  portion  of  the 
Z-plane  which,  after  the  preceding  explanation,  is  bounded  by  two  circles 

2iri  2irf 

with  the  origin  for  common  centre  and  of  radii  |  e w  "  |  and  |  e u  '  ;  and  the 
variations  of  the  function  within  the  rectangle  are  given  by  the  variations  of 
a  transformed  function  within  the  circular  ring.  The  characteristics  of  the 
one  function  at  points  in  the  rectangle  are  the  same  as  the  characteristics  of 
the  other  at  points  in  the  circular  ring :  and  therefore,  from  the  character 
of  the  assigned  function,  the  transformed  function  has  no  singularities  and  it 


214  FOURIER'S  THEOREM  [112. 

is  uniform  within  the  circular  ring.     Hence,  by  Laurent's  Theorem  (§  28), 
the  transformed  function  is  expressible  in  the  form 


a  series  which  converges  within  the  ring :  and  the  value  of  the  coefficient  an 

is  given  by 

1 


tvfj   Zn+* 

taken  along  any  circle  in  the  ring  concentric  with  the  boundaries. 

Retransforming  to  the  variable  z,  the  expression  for  the  original  function 
is 

71  =  +  oo          Zrnriz 

f(z)  =     2    ane~^~ . 

71=  -00 

The  series  converges  for  points  within  the  rectangle  and  therefore,  as  it 
is  periodic,  it  converges  within  the  portion  of  the  plane  assigned.  And  the 
value  of  an  is 

Zniriz 
(?\P       *»~ d? 

\z)  6          az, 


taken  along  a  path  which  is  the  equivalent  of  any  circle  in  the  ring  concentric 
with  the  boundaries,  that  is,  along  any  line  a'c'  perpendicular  to  the  lines 
which  bound  the  assigned  portion  of  the  plane. 

The  expression  of  the  function  can  evidently  be  changed  into  the  form 

Znvi,  _ 


1    r 

-± 

Wj    7 


where  the  integral  is  taken  along  the  piece  of  a  line,  perpendicular  to  the 
boundaries  and  intercepted  between  them. 

If  one  of  the  boundaries  of  the  portion  of  the  plane  be  at  infinity,  (so  that 
the  periodic  function  has  no  singularities  within  one  part  of  the  plane),  then 
the  corresponding  portion  of  the  ^-plane  is  either  the  part  within  or  the  part 
without  a  circle,  centre  the  origin,  according  as  the  one  or  the  other  of  the 
boundaries  is  at  oo  .  In  the  former  case,  the  terms  with  negative  indices 
n  are  absent  ;  in  the  latter,  the  terms  with  positive  indices  are  absent. 

113.  On  account  of  the  consequences  of  the  relation  subsisting  between 
the  variables  z  and  Z,  many  of  the  propositions  relating  to  general  uniform 
functions,  as  well  as  of  those  relating  to  multiform  functions,  can  be  changed, 
merely  by  the  transformation  of  the  variables,  into  propositions  relating  to 
simply-periodic  functions.  One  such  proposition  occurs  in  the  preceding 
section  ;  the  following  are  a  few  others,  the  full  development  being  unnecess 
ary  here,  in  consequence  of  the  foregoing  remark.  *  The  band  of  reference 
for  the  simply-periodic  functions  considered  will  be  supposed  to  include  the 


113.]  SIMPLY-PERIODIC   FUNCTIONS  215 

origin  :  and,  when  any  point  is  spoken  of,  it  is  that  one  of  the  series  of 
homologous  points  in  the  plane,  which  lies  in  the  band. 

We  know  that,  if  a  uniform  function  of  Z  have  no  essential  singularity, 
then  it  is  a  rational  algebraical  function,  which  is  integral  if  z  =  cc  be  the 
only  accidental  singularity  and  is  meromorphic  if  there  be  accidental  singu 
larities  in  the  finite  part  of  the  plane ;  and  every  such  function  has  as  many 
zeros  as  it  has  accidental  singularities. 

Hence  a  uniform  simply-periodic  function  with  z=cc  as  its  sole  essential 
singularity  has  as  many  zeros  as  it  has  infinities  in  each  band  of  the  plane ; 
the  number  of  points  at  which  it  assumes  a  given  value  is  equal  to  the  number 
of  its  zeros ;  and,  if  the  period  be  w,  the  function  is  a  rational  algebraical 

ZTTIZ 

function  of  e  a  ,  which  is  integral  if  all  the  singularities  be  at  an  infinite 
distance  and  is  meromorphic  if  some  (or  all)  of  them  be  in  a  finite  part  of 
the  plane.  But  any  number  of  the  zeros  and  any  number  of  the  infinities 
may  be  absorbed  in  the  essential  singularity  at  z  =  oo  . 

The  simplest  function  of  Z,  thus  restricted  to  have  the  same  number  of 
zeros  as  of  infinities,  is  one  which  has  a  single  zero  and  a  single  infinity  in 
the  finite  part  of  the  plane  ;  the  possession  of  a  single  zero  and  a  single  infinity 
will  therefore  characterise  the  most  elementary  simply-periodic  function. 
Now,  bearing  in  mind  the  relation 

Zniz 

Z=e<*, 

the  simplest  £-pomt  to  choose  for  a  zero  is  the  origin,  so  that  Z  =  1 ;  and  then 
the  simplest  ^-point  to  choose  for  an  infinity  at  a  finite  distance  is  \w,  (being 
half  the  period),  so  that  Z—  —  \.  The  expression  of  the  function  in  the 
Z-plane  with  1  for  a  zero  and  —  1  for  an  accidental  singularity  is 

Z~l 


and  therefore  assuming  as  the  most  elementary  simply-periodic  function  that 
which  in  the  plane  has  a  series  of  zeros  and  a  series  of  accidental  singularities 
all  of  the  first  order,  the  points  of  the  one  being  midway  between  those  of  the 
other,  its  expression  is 


A 


2iriz 

e"  -I 


Zniz 

which  is  a  constant  multiple  of  tan  — .     Since  e  "     is  a    rational  fractional 

CD 

function  of  tan  — ,  part  of  the  foregoing  theorem  can  be  re-stated  as  follows: — 

If  the  period  of  the  function  be  o>,  the  function  is  a  rational  algebraical 

function  of  tan  —  . 

n 


216  SIMPLY-PERIODIC  [113. 

Moreover,  in  the  general  theory  of  uniform  functions,  it  was  found  con 
venient  to  have  a  simple  element  for  the  construction  of  products,  there 
(§  53)  called  a  primary  factor:  it  was  of  the  type 


^Z-u 

where  the  function  G  ( -~ j  could  be  a  constant;  and  it  had  only  one  infinity 

and  one  zero. 

Hence  for  simply-periodic  functions  we  may  regard  tan  —  as  a  typical 

primary  factor  when  the  number  of  irreducible  zeros  and  the  (equal)  number 
of  irreducible  accidental  singularities  are  finite.  If  these  numbers  should 
tend  to  an  infinite  limit,  then  an  exponential  factor  might  have  to  be 

associated  with  tan  — ;  and  the  function  in  that  case  might  have  essential 
singularities  elsewhere  than  at  z  =  oo  . 

114.  We  can  now  prove  that  every  uniform  function,  which  has  no 
essential  singularities  in  the  finite  part  of  the  plane  and  is  such  that  all  its 
accidental  singularities  and  its  zeros  are  arranged  in  groups  equal  and 
finite  in  number  at  equal  distances  along  directions  parallel  to  a  given 
direction,  is  a  simply-periodic  function. 

Let  to  be  the  common  period  of  the  groups  of  zeros  and  of  singularities : 
and  let  the  plane  be  divided  into  bands  by  parallel  lines,  perpendicular  to 
any  line  representing  w.  Let  a,  b,  ...  be  the  zeros,  a,  /3,  ...  the  singularities 
in  any  one  band. 

Take  a  uniform  function  </>  (z),  simply-periodic  in  <w  and  having  a  single 
zero  and  a  single  singularity  in  the  band :  we  might  take  tan  —  as  a  value 
of  <f>  (z).  Then 


is  a  simply-periodic  function  having  only  a  single  zero,  viz.,  z  =  a  and  a  single 
singularity,  viz.,  z  —  a.  ;  for  as  <f>  {z}  has  only  a  single  zero,  there  is  only  a  single 
point  for  which  (f>(z)  =  <f)  (a),  and  a  single  point  for  which  <£  (z)  —  $  (a).  Hence 


is  a  simply-periodic  function  with  all  the  zeros  and  with  all  the  infinities  of 
the  given  function  within  the  band.  But  on  account  of  its  periodicity  it  has 
all  the  zeros  and  all  the  infinities  of  the  given  function  over  the  whole  plane  ; 
hence  its  quotient  by  the  given  function  has  no  zero  and  no  singularity  over 
the  whole  plane  and  therefore  it  is  a  constant  ;  that  is,  the  given  function, 


114.]  FUNCTIONS  217 

save  as  to  a  constant  factor,  can  be  expressed  in  the  foregoing  form.     It  is 
thus  a  simply-periodic  function. 

This  method  can  evidently  be  used  to  construct  simply-periodic  functions,  having 
assigned  zeros  and  assigned  singularities.  Thus  if  a  function  have  a  +  mat  as  its  zeros  and 
c+m'<o  as  its  singularities,  where  m  and  m'  have  all  integral  values  from  —  oo  to  +00, 
the  simplest  form  is  obtained  by  taking  a  constant  multiple  of 

TTZ  7T« 

tan tan  — 


TTZ      ,       TTC 
tan tan  — 


Ex.     Construct  a  function,  simply-periodic  in  w,  having  zeros  given  by  (m+^)o>  and 

)o>  and  singularities  by  (m  +  i)co  and  (m  +  §)  co. 
The  irreducible  zeros  are  ^co  and  f  w  ;  the  irreducible  singularities  are  \u>  and  §«.     Now 


f.TTZ  \     (  TTZ  ,     \ 

I  tan tan  ATT  I  I  tan tan  |TT  I 

/  \         <"  /    \         M  / 

7TZ  \     (  TTZ  „     \ 

tan tan  JTT  ]  ( tan tan  |TT  I 

/    \  / 


is  evidently  a  function,  initially  satisfying  the  required  conditions.  But,  as  tari^r  is 
infinite,  we  divide  out  by  it  and  absorb  it  into  A'  as  a  factor ;  the  function  then  takes 
the  form 

1  +  tan  - 


3-tan'7^ 

60 

We  shall  not  consider  simply-periodic  functions,  which  have  essential 
singularities  elsewhere  than  at  z  =  <x> ;  adequate  investigation  will  be  found 
in  the  second  part  of  Guichard's  memoir,  (I.e.,  p.  147).  But  before  leaving  the 
consideration  of  the  present  class  of  functions,  one  remark  may  be  made.  It 
was  proved,  in  our  earlier  investigations,  that  uniform  functions  can  be 
expressed  as  infinite  series  of  functions  of  the  variable  and  also  as  infinite 
products  of  functions  of  the  variable.  This  general  result  is  true  when  the 
functions  in  the  series  and  in  the  products  are  simply-periodic  in  the  same 
period.  But  the  function,  so  represented,  though  periodic  in  that  common 
period,  may  also  have  another  period :  and,  in  fact,  many  doubly-periodic 
functions  of  different  kinds  (§  136)  are  often  conveniently  expressed  as  infinite 
converging  series  or  infinite  converging  products  of  simply-periodic  functions. 

Any  detailed  illustration  of  this  remark  belongs  to  the  theory  of  elliptic  functions :  one 
simple  example  must  suffice. 

,  ima' 

Let  the  real  part  of  -  -  be  negative,  and  let  q  denote  e  "  ;  then  the  function 


being  an  infinite  converging  series  of  powers  of  the  simply-periodic  function  e  "    ,  is  finite 
everywhere  in  the  plane.     Evidently  6  (z)  is  periodic  in  o>,  so  that 

=  6  (z). 


218  DOUBLY-PERIODIC  [114. 


° 


Again,  0(s  +  «»')  =    2 


the  change  in  the  summation  so  as  to  give  $  (z)  being  permissible  because  the  extreme 
terms  for  the  infinite  values  of  n  can  be  neglected  on  account  of  the  assumption  with 
regard  to  q.  There  is  thus  a  pseudo-periodicity  for  6(z)  in  a  period  <•>'. 


Similarly,  if  0s(z)=  q*  e 


2J7TZ 

63(z  +  <a')  =  -e     "    6(z). 

Then  63(z)  -r-d(z)  is  doubly-periodic  in  w  and  2co',  though  constructed  only  from 
functions  simply-periodic  in  w  :  it  is  a  function  with  an  infinite  number  of  irreducible 
accidental  singularities  in  a  band. 

115.  We  now  pass  to  doubly-periodic  functions  of  a  single  variable,  the 
periodicity  being  additive.  The  properties,  characteristic  of  this  important 
class  of  functions,  will  be  given  in  the  form  either  of  new  theorems  or 
appropriate  modifications  of  theorems,  already  established  ;  and  the  develop 
ment  adopted  will  follow,  in  a  general  manner,  the  theory  given  by  Liouville*. 
It  will  be  assumed  that  the  functions  are  uniform,  unless  multiformity  be 
explicitly  stated,  and  that  all  the  singularities  in  the  finite  part  of  the  plane 
are  accidental  "f*. 

The  geometrical  representation  of  double-periodicity,  explained  in  §  105, 
will  be  used  concurrently  with  the  analysis;  and  the  parallelogram  of 
periods,  to  which  the  variable  argument  of  the  function  is  referred,  is  a 
fundamental  parallelogram  (§  109)  with  periods  J  2co  and  2&>'.  An  angular 
point  £0  for  the  parallelogram  of  reference  can  be  chosen  so  that  neither  a 
zero  nor  a  pole  of  the  function  lies  on  the  perimeter;  for  the  number 
of  zeros  and  the  number  of  poles  in  any  finite  area  must  be  finite, 
otherwise  they  would  form  a  continuous  line  or  a  continuous  area,  or  thej 
would  be  in  the  vicinity  of  an  essential  singularity.  This  choice  will,  ir 

*  In  his  lectures  of  1847,  edited  by  Borchardt  and  published  in  Crelle,  t.  Ixxxviii,  (1880),  pp. 
277  —  310.     They  are  the  basis   of  the   researches   of  Briot  and  Bouquet,  the  most  complet 
exposition   of  which  will  be  found  in   their   Theorie  des  fonctions   elliptiques,  (2nd  ed.),  pp. 
239—280. 

t  For  doubly-periodic  functions,  which  have  essential  singularities,  reference  should  be  made 
to  Guichard's  memoir,  (the  introductory  remarks  aud  the  third  part),  already  quoted  on  p.  147,  note. 

J  The  factor  2  is  introduced  merely  for  the  sake  of  convenience. 


115.]  FUNCTIONS  219 

general,  be  made ;  but,  in  particular  cases,  it  is  convenient  to  have  the  origin 
as  an  angular  point  of  the  parallelogram  and  then  it  not  infrequently  occurs 
that  a  zero  or  a  pole  lies  on  a  side  or  at  a  corner.  If  such  a  point  lie  on  a  side, 
the  homologous  point  on  the  opposite  side  is  assigned  to  the  parallelogram 
which  has  that  opposite  side  as  homologous;  and  if  it  be  at  an  angular  point, 
the  remaining  angular  points  are  assigned  to  the  parallelograms  which  have 
them  as  homologous  corners. 

The  parallelogram  of  reference  will  therefore,  in  general,  have  z0,  z0  +  2&>, 
z0  +  2&/,  z0  +  2&>  +  2&>'  for  its  angular  points  ;  but  occasionally  it  is  desirable 
to  .take  an  equivalent  parallelogram  having  z0  ±  &>  +  &>'  as  its  angular 
points. 

When  the  function  is  denoted  by  </>  (2),  the  equations  indicating  the 
periodicity  are 

<£  (z  +  2<w)  =  (f>  (z)  =  (f>  (z  +  2&/). 

116.  We  now  proceed  to  the  fundamental  propositions  relating  to 
doubly-periodic  functions. 

I.  Every  doubly -periodic  function  must  have  zeros  and  infinities  within 
the  fundamental  parallelogram. 

For  the  function,  not  being  a  constant,  has  zeros  somewhere  in  the  plane 
and  it  has  infinities  somewhere  in  the  plane ;  and,  being  doubly-periodic,  it 
experiences  within  the  parallelogram  all  the  variations  that  it  can  have  over 
the  plane. 

COROLLARY.     The  function  cannot  be  a  rational  integral  function  of  z. 

For  within  a  parallelogram  of  finite  dimensions  an  integral  function  has 
no  infinities  and  therefore  cannot  represent  a  doubly-periodic  function. 

An  analytical  form  for  <j)  (z)  can  be  obtained  which  will  put  its  singu 
larities  in  evidence.  Let  a  be  such  a  pole,  of  multiplicity  n ;  then  we  know 
that,  as  the  function  is  uniform,  coefficients  A  can  be  determined  so  that  the 
function 

f(*    ~  (z-a)n~  (z-a)n-1~'"~(z-a)2~  z^a 

is  finite  in  the  vicinity  of  a ;  but  the  remaining  poles  of  <j>  (z)  are  singularities 
of  this  modified  function.  Proceeding  similarly  with  the  other  singularities 
b,  c,...,  which  are  finite  in  number  and  each  of  which  is  finite  in  degree,  we 
have  coefficients  A,  B,  C,...  determined  so  that 

A^<        V       i?        K' 

9  (z)  —    2,      f  Z  T r 

is  finite  in  the  vicinity  of  every  pole  of  <f)  (z)  within  the  parallelogram  and 
therefore  is  finite  everywhere  within  the  parallelogram.  Let  its  value  be 


220  PROPERTIES  [116. 

%(X);  then  for  points  lying  within  the  parallelogram,  the  function  <f>(z)  is 
expressed  in  the  form 


+  A*  + 

^             1                  1 

A, 

T                 T  ; 

2  —  a     ( 

ft 

+       1     i 

\9      1       '  '  '        '       / 

z  -  a>              ( 
B2 

z  -  a)n 
Bm 

X.     '     / 

£—6        ( 

7   \  0       '       •  •  •       1       / 

z-b)m 

H, 

_L                      _L 

#2 

S±i 

z-h^  (z-h?  '         r  (z-h)1' 

But  though  <£  (^)  is  periodic,  ^  (2?)  is  not  periodic.  It  has  the  property  of 
being  finite  everywhere  within  the  parallelogram ;  if  it  were  periodic,  it 
would  be  finite  everywhere,  and  therefore  could  have  only  a  constant  value ; 
and  then  <f>  (z)  would  be  an  algebraical  meromorphic  function,  which  is  not 
periodic.  The  sum  of  the  fractions  in  $  (z)  may  be  called  the  fractional 
part  of  the  function :  owing  to  the  meromorphic  character  of  the  function, 
it  cannot  be  evanescent. 

The  analytical  expression  can  be  put  in  the  form 
(z  -  a)~n  (z  -  6)-™. .  .(z  -  h)~l  F(z\ 

where  F(z)  is  finite  everywhere  within  the  parallelogram.  If  a,  /3,  ...,  ij  be 
all  the  zeros,  of  degrees  v,  p,  ...,  X,  within  the  parallelogram,  then 

F(z)  =  (z-a)v(z-py  ...(z-^G(z\ 

where  G  (z)  has  no  zero  within  the  parallelogram ;  and  so  the  function  can 
be  expressed  in  the  form 

(z-a)n(z-b}m...(z-h)1  G^' 

where  G  (z)  has  no  zero  and  no  infinity  within  the  parallelogram  or  on  its 
boundary ;  and  G  {z)  is  not  periodic. 

The  order  of  a  doubly-periodic  function  is  the  sum  of  the  multiplicities 
of  all  the  poles  which  the  function  has  within  a  fundamental  parallelogram; 
and,  the  sum  being  n,  the  function  is  said  to  be  of  the  nth  order.  All 
these  singularities  are,  as  already  remarked,  accidental;  it  is  convenient 
to  speak  of  any  particular  singularity  as  simple,  double,  . . .  according  to  its 
multiplicity. 

If  two  doubly-periodic  functions  u  and  v  be  such  that  an  equation 


is  satisfied  for  constant  values  of  A,  B,  C,  the  functions  are  said  to  be 
equivalent  to  one  another.  Equivalent  functions  evidently  have  the  same 
accidental  singularities  in  the  same  multiplicity. 

II.     The  integral  of  a  doubly-periodic  function  round  the  boundary  of  a 
fundamental  parallelogram  is  zero. 


116.] 


OF   DOUBLY-PERIODIC   FUNCTIONS 


221 


Let  ABCD  be  a  fundamental  parallelogram,  the  boundary  of  it  being 
taken  so  as  to  pass  through  no  pole  of  the 
function.     Let  A    be  z0,  B  be   z0+2ca,  and*  <= 

D  be  z0  +  2a)':    then  any  point  in  AB  is  /  ° 

/Q*  Q, 


where  £  is  a  real  quantity  lying  between  0  and  1 ; 
and  therefore  the  integral  along  AB  is 

rl 


Any  point  in  EG  is  z0  +  2<w  +  2&>'£,  where  £  is  a  real  quantity  lying  between  0 
and  1  ;  therefore  the  integral  along  BC  is 


(o  'dt, 

o 

since  <^>  is  periodic  in  2&). 

Any  point   in   DC  is  s  +  2o>'  +  2<wZ,  where  <  is  a   real   quantity  lying 
between  0  and  1  ;  therefore  the  integral  along  CD  is 


f° 

J  1 


2ft)' 


=  -  I 

J  o 

Similarly,  the  integral  along  DA  is 

=  -  I   cf>  Oo  +  2o>'«)  2w'^. 

J  o 

Hence  the  complete  value  of  the  integral,  taken  round  the  parallelogram,  i 

fi 
=      <j>(z0 

Jo 


which  ^  is   manifestly   zero,   since   each   of  the  integrals  is  the  integral  of 
a  continuous  function. 

COROLLARY.     Let  ty(z)  be  any  uniform  function   of  zt  not   necessarily 
doubly-periodic,   but   without   singularities    on   the    boundary.      Then    the 

*  The  figure  implies  that  the  argument  of  w'  is  greater  than  the  argument  of  w,  a 
hypothesis  which,  though  unimportant  for  the  present  proposition,  must  be  taken  account  of 
hereafter  (e.g.,  §  129). 


222  INTEGRAL   RESIDUE  [116. 


integral  jty  (z)  dz  taken  round  the  parallelogram  of  periods  is  easily  seen 
to  be 


n  ri 

•^  (z(}  +  Scot)  2udt  +  I   ^(z0  +  2a>  +  2m't)  2a>'dt 

Jo  J  o 

ri  ri 

-      V  (*o  +  2o>'  +  2a>t)  2(odt  -      ^  (z0  +  2to't)  2w'dt  ; 

Jo  Jo 


or,  if  we  write 


/•  ri  ri 

then  U-  (2)  ^  =  I   I/TJ  (>0  +  2w't)  2m  dt  -      ^  (z,  +  2wt)  2(odt, 

J  Jo  Jo 

where  on  the  left-hand  side  the  integral  is  taken  positively  round  the 
boundary  of  the  parallelogram  and  on  the  right-hand  side  the  variable  t 
in  the  integrals  is  real. 

The  result  may  also  be  written  in  the  form 

r  rD  rx 

\-^r(z)dz=\    ^  (z)  dz  —  I    -»K  (z)  dz, 

J  J  A  J  A 

the  integrals  on  the  right-hand  side  being  taken  along  the  straight  lines  AD 
and  AB  respectively. 

Evidently  the  foregoing  main  proposition  is  established,  when  -^  (£)  and 
T/r2  (f)  vanish  for  all  values  of  £. 

III.  If  a  doubly  -periodic  function  $(z)  have  infinities  Oj,  a2,  ...  within 
the  parallelogram,  and  if  Al,  A2,  ...  be  the  coefficients  of  (z  —  e^)"1,  (z  —  a^r1,  .  .  . 
respectively  in  the  fractional  part  of  (j>  (z)  when  it  is  expanded  in  the  parallelo 
gram,  then 

A1  +  A2+...=0. 

As  the  function  <f>(z)  is  uniform,  the  integral  f(f>(z)dz  is,  by  (§  19,  II.),  the 
sum  of  the  integrals  round  a  number  of  curves  each  including  one  and  only 
one  of  the  infinities  within  that  parallelogram. 

Taking  the  expression  for  (f>(z)  on  p.  220,  the  integral  Amf(z  —  a)~mdz 
round  the  curve  enclosing  a  is  0,  if  m  be  not  unity,  and  is  Z>jriAl,  if  m  be 
unity;  the  integral  Kmf(z  —  k)~mdz  round  that  curve  is  0  for  all  values  of  m 
and  for  all  points  k  other  than  a  ;  and  the  integral  /^  (z)  dz  round  the  curve 
is  zero,  since  %  (z)  is  uniform  and  finite  everywhere  in  the  vicinity  of  a.  Hence 
the  integral  of  <£  (z)  round  a  curve  enclosing  c^  alone  of  all  the  infinities  is 


Similarly  the  integral  round  a  curve  enclosing  a.2  alone  is  27riA.2;  and  so 
on,  for  each  of  the  curves  in  succession. 

Hence  the  value  of  the  integral  round  the  parallelogram  is 

2-rnZA. 


116.]  OF   FUNCTIONS   OF   THE   SECOND   ORDER  223 

But  by  the  preceding  proposition,  the  value  of  /(/>  (2)  dz  round  the  parallelo 
gram  is  zero  ;  and  therefore 


This  result  can  be  expressed  in  the  form  that  the  sum  of  the  residues*  of  a 
doubly  -periodic  function  relative  to  a  fundamental  parallelogram  of  periods 
is  zero. 

COROLLARY  1.     A  doubly-periodic  function  of  the  first  order  does  not 

exist. 

Let  such  a  function  have  a  for  its  single  simple  infinity.  Then  an 
expression  for  the  function  within  the  parallelogram  is 

A 

^-a  +  *^> 

where  ^  (2)  is  everywhere  finite  in  the  parallelogram.  By  the  above  propo 
sition,  A  vanishes  ;  and  so  the  function  has  no  infinity  in  the  parallelogram. 
It  therefore  has  no  infinity  anywhere  in  the  plane,  and  so  is  merely  a 
constant  :  that  is,  qua  function  of  a  variable,  it  does  not  exist. 

COROLLARY  2.  Doubly-periodic  functions  of  the  second  order  are  of  two 
classes. 

As  the  function  is  of  the  second  order,  the  sum  of  the  degrees  of  the 
infinities  is  two.  There  may  thus  be  either  a  single  infinity  of  the  second 
degree  or  two  simple  infinities. 

In  the  former  case,  the  analytical  expression  of  the  function  is 


where  a  is  the  infinity  of  the  second  degree  and  ^  (z)  is  holomorphic  within 
the  parallelogram.  But,  by  the  preceding  proposition,  A1  =  0;  hence  the 
analytical  expression  for  a  doubly-periodic  function  with  a  single  irreducible 
infinity  a  of  the  second  degree  is 


(z  -  of  T  *  v 

within  the  parallelogram.     Such  functions  of  the  second  order,  which  have 
only  a  single  irreducible  infinity,  may  be  called  the  first  class. 
In  the  latter  case,  the  analytical  expression  of  the  function  is 


where  c,  and  c2  are  the  two  simple  infinities  and  x(z}  ig  finite  within  the 
parallelogram.     Then 


See  p.  42. 


224  PROPERTIES   OF   FUNCTIONS  [116. 

so   that,  if   Cl  =  -  C.2  =  C,   the   analytical  expression   for   a   doubly-periodic 
function  with  two  simple  irreducible  infinities  a1  and  «2  ig 


n 
G 


(     1  1 

(  - 

\z-a-L     z  - 


within  the  parallelogram.  Such  functions  of  the  second  order,  which  have 
two  irreducible  infinities,  may  be  called  the  second  class. 

COROLLARY  3.  If  within  any  parallelogram  of  periods  a  function  is 
only  of  the  second  order,  the  parallelogram  is  fundamental. 

COROLLARY  4.  A  similar  division  of  doubly -periodic  functions  of  any 
order  into  classes  can  be  effected  according  to  the  variety  in  the  constitution  of 
the  order,  the  number  of  classes  being  the  number  of  partitions  of  the  order. 

The  simplest  class  of  functions  of  the  nth  order  is  that  in  which  the 
functions  have  only  a  single  irreducible  infinity  of  the  nth  degree.  Evi 
dently  the  analytical  expression  of  the  function  within  the  parallelogram  is 

G,  G,  Gn 

(z  -  a)2     (z  -  a)3  (z  -  a)n     *  ^  '' 

where  ^  (z)  is  holomorphic  within  the  parallelogram.  Some  of  the  coefficients 
G  may  vanish ;  but  all  may  not  vanish,  for  the  function  would  then  be  finite 
everywhere  in  the  parallelogram. 

It  will  however  be  seen,  from  the  next  succeeding  propositions,  that  the 
division  into  classes  is  of  most  importance  for  functions  of  the  second 
jrder. 

IV.  Two  functions,  which  are  doubly-periodic  in  the  same  periods*,  and 
which  have  the  same  zeros  and  the  same  infinities  each  in  the  same  degrees 
respectively,  are  in  a  constant  ratio. 

Let  <f)  and  ^  be  the  functions,  having  the  same  periods;  and  let  a  of 
degree  v,  /3  of  degree  fi,  ...  be  all  the  irreducible  zeros  of  <£  and  T/T;  arid  a  of 
degree  n,  b  of  degree  m,  ...  be  all  the  irreducible  infinities  of  <f>  and  of  ty. 
Then  a  function  G  (z),  without  zeros  or  infinities  within  the  parallelogram, 
exists  such  that 

, , ,  =  (z-a)v(z-py  ...  G     _ 

and  another  function  H(z),  without  zeros  or  infinities  within  the  parallelo 
gram,  exists  such  that 


Hence  *(*)_<?(*) 

- 


Now  the  function  on  the  right-hand  side  has  no  zeros  in  the  parallelogram, 
for  G  has  no  zeros  and  H  has  no  infinities  ;  and  it  has  no  infinities  in  the 

*  Such  functions  will  be  called  homoperiodic. 


116.]  OF   THE   SECOND   ORDER  225 

parallelogram,  for  G  has  no  infinities  and  H  has  no  zeros  :  hence  it  has 
neither  zeros  nor  infinities  in  the  parallelogram.  Since  it  is  equal  to  the 
function  on  the  left-hand  side,  which  is  a  doubly-periodic  function,  it  has  no 
zeros  and  no  infinities  in  the  whole  plane  ;  it  is  therefore  a  constant,  say 
A.  Thus* 


V.     Two  functions  of  the  second  order,  doubly  -periodic  in  the  same  periods 
and  having  the  same  infinities,  are  equivalent  to  one  another. 

If  one  of  the  functions  be  of  the  first  class  in  the  second  order,  it  has  one 
irreducible  double  infinity,  say  at  a  ;  so  that  we  have 


where  %(z)  is  finite  everywhere  within  the  parallelogram.  Then  the  other 
function  also  has  z  =  a  for  its  sole  irreducible  infinity  and  that  infinity  is  of 
the  second  degree  ;  therefore  we  have 

TT 


where  ^  (z)  is  finite  everywhere  within  the  parallelogram.     Hence 


Now  x  and  %x  are  finite  everywhere  within  the  parallelogram,  and  therefore 
so  is  H%  —  Gfo.  But  H%  —  Gfo,  being  equal  to  the  doubly-periodic  function 
H(j)  —  Gijr,  is  therefore  doubly-periodic  ;  as  it  has  no  infinities  within  the 
parallelogram,  it  consequently  can  have  none  over  the  plane  and  therefore  it 
is  a  constant,  say  7.  Thus 


proving  that  the  functions  <j>  and  ty  are  equivalent. 

If  on  the  other  hand  one  of  the  functions  be  of  the  second  class  in  the 
second  order,  it  has  two  irreducible  simple  infinities,  say  at  6  and  c,  so  that 
we  have 


where  6(z)  is  finite  everywhere  within  the  parallelogram.  Then  the  other 
function  also  has  z  =  b  and  z  =  c  for  its  irreducible  infinities,  each  of  them 
being  simple  ;  therefore  we  have 


where  6l  (z)  is  finite  everywhere  within  the  parallelogram.     Hence 

(z)  -  Cty  (z)  =  De  (z)  -  Cei  (z}. 


*  This  proposition  is  the  modified  form  of  the  proposition  of  §  52,  when  the  generalising 
exponential  factor  has  been  determined  so  as  to  admit  of  the  periodicity. 

F.  15 


226  IRREDUCIBLE   ZEROS  [116. 

The  right-hand  side,  being  finite  everywhere  in  the  parallelogram,  and  equal 
to  the  left-hand  side  which  is  a  doubly-periodic  function,  is  finite  everywhere 
in  the  plane  ;  it  is  therefore  a  constant,  say  B,  so  that 


proving  that  <£  and  ty  are  equivalent  to  one  another. 

It  thus  appears  that  in  considering  doubly-periodic  functions  of  the  second 
order,  homoperiodic  functions  of  the  same  class  are  equivalent  to  one  another 
if  they  have  the  same  infinities  ;  so  that,  practically,  it  is  by  their  infinities 
that  homoperiodic  functions  of  the  second  order  and  the  same  class  are  dis 
criminated. 

COROLLARY  1.  If  two  equivalent  functions  of  tlie  second  order  have  one 
zero  the  same,  all  their  zeros  are  the  same. 

For  in  the  one  class  the  constant  /,  and  in  the  other  class  the  constant  B, 
is  seen  to  vanish  on  substituting  for  z  the  common  zero  ;  and  then  the  two 
functions  always  vanish  together. 

COROLLARY  2.  If  two  functions,  doubly-periodic  in  the  same  periods  but 
not  necessarily  of  the  second  order,  have  the  same  infinities  occurring  in  such  a,  j 
way  that  the  fractional  parts  of  the  two  functions  are  the  same  except  as  to  a 
constant  factor,  the  functions  are  equivalent  to  one  another.  And  if,  in 
addition,  they  have  one  zero  common,  then  all  their  zeros  are  common,  so 
that  the  functions  are  then  in  a  constant  ratio. 

COROLLARY  3.  If  two  functions  of  the  second  order,  doubly-periodic  in( 
the  same  periods,  have  their  zeros  the  same,  and  one  infinity  common,  they  are  ^ 
in  a  constant  ratio. 

VI.  Every  doubly  -periodic  function  has  as  many  irreducible  zeros  as  it 
has  irreducible  infinities. 

Let  <£  (z)  be  such  a  function.     Then 


z  +h  —  z 

is  a  doubly-periodic  function  for  any  value  of  h,  for  the  numerator  is  doubly- 
periodic  and  the  denominator  does  not  involve  z  ;  so  that,  in  the  limit  when 
h  =  0,  the  function  is  doubly-periodic,  that  is,  </>'  (z)  is  doubly-periodic. 

Now  suppose  <f>(z)  has  irreducible  zeros  of  degree  m1  at  a1}  ra2  at  a2,  ..., 
and  has  irreducible  infinities  of  degree  /^  at  «1}  yu,2  at  «2,  ...  ;  so  that  the 
number  of  irreducible  zeros  is  Wj  +  ra2  +  .  .  .  ,  and  the  number  of  irreducible 
infinities  is  ^1  +  /i2  +  ...,  both  of  these  numbers  being  finite.  It  has  been 
shewn  that  <£  {z)  can  be  expressed  in  the  form 


116.]  AND   IRREDUCIBLE   INFINITIES  227 

whore  F(z)  has  neither  a  zero  nor  an  infinity  within,  or  on  the  boundary  of, 
the  parallelogram  of  reference. 

Since  F(z)  has  a  value,  which  is  finite,  continuous  and  different  from  zero 

Tjlt   /     \ 

everywhere  within  the  parallelogram  or  on  its  boundary,  the  function  -p4-r 

*  W 
is  not  infinite  within  the  same  limits.     Hence  we  have 


rr  -    ~    —       ... 

9  (z)  z—a±     z  —  «2 

+  -*  +  =*.  +  .. 

z  —  ttj     z  —  a2 

where  g  (z)  has  no  infinities  within,  or  on  the  boundary  of,  the  parallelogram 
of  reference.  But,  because  <f>  (z)  and  <f>  (z)  are  doubly-periodic,  their  quotient 
is  also  doubly-periodic  ;  and  therefore,  applying  Prop.  II.,  we  have 

m^  +  w2  +  .  .  .  —  ^  —  p2  —  .  .  .  =  0, 
that  is,  m1+m2  +  ...  =  fj,!  +  fi2+  ..., 

or  the  number  of  irreducible  zeros  is  equal  to  the  number  of  irreducible 
infinities. 

COROLLARY  I.  The  number  of  irreducible  points  for  which  a  doubly  - 
periodic  function  assumes  a  given  value  is  equal  to  the  number  of  irreducible 
zeros. 

For  if  the  value  be  A,  every  infinity  of  $(z)  is  an  infinity  of  the  doubly- 
periodic  function  $  (z)  —  A  ;  hence  the  number  of  the  irreducible  zeros  of  the 
latter  is  equal  to  the  number  of  its  irreducible  infinities,  which  is  the  same  as 
the  number  for  <£  (z}  and  therefore  the  same  as  the  number  of  irreducible 
zeros  of  <£  (z).  And  every  irreducible  zero  of  <£  (z}  —  A  is  an  irreducible 
point,  for  which  <£  (z)  assumes  the  value  A. 

COROLLARY  II.  A  doubly-periodic  function  with  only  a  single  zero  does 
not  exist;  a  doubly  -periodic  function  of  the  second  order  has  two  zeros;  and, 
generally,  the  order  of  a  function  can  be  measured  by  its  number  of  irreducible 
zeros. 

Note.  It  may  here  be  remarked  that  the  doubly-periodic  functions 
(§  115),  that  have  only  accidental  singularities  in  the  finite  part  of  the 
plane,  have  z  =  oo  for  an  essential  singularity.  It  is  evident  that  for  infinite 
values  of  z,  the  finite  magnitude  of  the  parallelogram  of  periods  is  not 
recognisable  ;  and  thus  for  z  =  GO  the  function  can  have  any  value,  shewing 
that  z  =  oo  is  an  essential  singularity. 

VII.  Let  a1}  a2)...  be  the  irreducible  zeros  of  a  function  of  degrees 
w1;  m2,  ...  respectively  ;  a1}  «2,  ...  its  irreducible  infinities  of  degrees  /^,  /u,2,  ... 
respectively;  and  z1,z2,...  the  irreducible  points  where  it  assumes  a  value  c, 
which  is  neither  zero  nor  infinity,  their  degrees  being  M1}  M.2)  ...  respectively. 

15—2 


228  IRREDUCIBLE   ZEROS  [116. 

Then,  except  possibly  as  to  additive  multiples  of  Hie  periods,  the  quantities 
2  mrar,    2  UrCir  and    2  Mrzr  are  equal  to  one  another,  so  that 

r=l  r=l  r=l 

2  mrar  =  2  Mrzr  =  2  prctr  (mod.  2o>,  2&/)- 

r=l  r=l  r=l 

Let  (/>  (/)  be  the  function.  Then  the  quantities  which  occur  are  the  sums 
of  the  zeros,  the  assigned  values,  and  the  infinities,  the  degree  of  each  being 
taken  account  of  when  there  is  multiple  occurrence  ;  and  by  the  last 
proposition  these  degrees  satisfy  the  relations 


The  function  <f)(z)  —  c  is  doubly-periodic  in  2«u  and  2&>' ;  its  zeros  are 
z1}  z.2, ...  of  degrees  M1}  M^,...  respectively;  and  its  infinities  are  ctl,  «2, ...  of 
degrees  /i1}  yn2,  •••,  being  the  same  as  those  of  <£(Y).  Hence  there  exists  a 
function  G(z),  without  either  a  zero  or  an  infinity  lying  in  the  parallelogram 
or  on  its  boundary,  such  that  </>  0)  -  c  can  be  expressed  in  the  form 

^l*1C.(*I*a>!'" G  (*) 

for  all  points  not  outside  the  parallelogram ;  and  therefore,  for  points  in  that 

region 

<f>'(Y)         ^     Mr       ^     *r         G'(z) 

\      /~**  /     \    • 


/      \ 

<j)(z)  —  C       r=l  Z  —  Zr  Z—  O.r 

Hence 

z$(z)        ~    Mrz       v    prz       zG'  (z) 

.   .  >.  -  —    2<    -      ---  2*  ---  1  —  .~  ,  . 
$(z)  —  C        r=l  z  —  zr  Z—  ar         W  (*) 

=  2  Jfr+  2 


, 

~r 


*  ~r    /-v  /    -.     , 

=\Z—  Zr  Z—OLr         (r(z) 

2  Mr=  2  nr. 

r=l  r=l 

Integrate  both  sides  round  the  boundary  of  the  fundamental  parallelogram. 
Because  G  (z)  has  no  zero  and  no  infinity  in  the  included  region  and  does  not 
vanish  along  the  curve,  the  integral 

'zG'(z) 


I 


dz 


G(z) 

vanishes.     But  the  points  z{  and  04  are  enclosed  in  the  area  ;  and  therefore 
the  value  of  the  right-hand  side  is 

2iri  2  Mrzr  —  Ziri  2  /V*r, 


so 


that 


\Z)  —  c 
the  integral  being  extended  round  the  parallelogram. 


116.]  AND   IRREDUCIBLE    INFINITIES  229 

zd>'  (z) 

Denoting  the  subject  of  integration      ,  by/(^),  we  have 

<p(z)  —  c 

-/«=*"         - 


and   therefore,  by  the    Corollary  to  Prop.  II.,  the   value  of  the    foregoing 
integral  is 

*•  r  £¥-*-**  r  £¥-*• 

JA<f>(Z)-C  JA(j)(z)-G 

the  integrals  being  taken  along  the  straight  lines  AD  and  AB  respectively 
(fig.  33,  p.  221). 

Let  w  —  <f)(z)  —  c;  then,  as  z  describes  a  path,  w  will  also  describe  a  single 
path  as  it  is  a  uniform  function  of  z.  When  z  moves  from  A  to  D,  w  moves 
from  (j>(A)-c  by  some  path  to  (f>(D)  —  c,  that  is,  it  returns  to  its  initial 
position  since  <f>  (D)  =  <f>  (A)  ;  hence,  as  z  describes  AD,  w  describes  a  simple 
closed  path,  the  area  included  by  which  may  or  may  not  contain  zeros  and 
infinities  of  w.  Now 

dw  =  <f>'  (z)  dz, 

CD      <£'  (z\ 

and  therefore  the  integral  I      ,,\        dz  is  equal  to 
*       JAJ>(*)-C 


I 


dw 
w 


taken  in  some  direction  round  the  corresponding  closed  path  for  w.  This 
integral  vanishes,  if  no  w-zero  or  w-infinity  be  included  within  the  area 
bounded  by  the  path  ;  it  is  +  Im'iri,  if  m  be  the  excess  of  the  number  of 
included  zeros  over  the  number  of  included  infinities,  the  +  or  —  sign  being 
taken  with  a  positive  or  a  negative  description  ;  hence  we  have 


where  m  is  some  positive  or  negative  integer  and  may  be  zero.     Similarly 


where  n  is  some  positive  or  negative  integer  and  may  be  zero. 

Thus  27Ti  (2,MrZr  ~  2/V*,-)  =  2w  .  2w7n  —  2a)'  .  Smri, 

and  therefore  ^Mrzr  —  ^prir  =  2ma>  —  2?io>' 

=  0  (mod.  2&),  2o>'). 

Finally,  since  ^Mrzr  =  2/v*r  whatever  be  the  value  of  c,  for  the  right-hand 


230  DOUBLY-PERIODIC    FUNCTIONS  [116. 

side  is  independent  of  c,  we  may  assign  to  c  any  value  we  please.  Let  the 
value  zero  be  assigned ;  then  ^Mrzr  becomes  Smrar,  so  that 

^mrar  =  "2/j,rf*r  (mod.  2&>,  2&/). 

The  combination  of  these  results  leads  to  the  required  theorem*,  expressed 
by  the  congruences 

2  mrar  =  2  Mrzr  =  2  ^r^r  (mod.  2o>,  2&>'). 

r=l  r=l  r=l 

Note.  Any  point  within  the  parallelogram  can  be  represented  in  the 
form  z0  +  a2&>  +  62&>',  where  a  and  6  are  real  positive  quantities  less  than 

unity.     Hence 

2  Mrzr  =  Az'2a>  +  Bz2a>/  +  z£Mr, 

where  J.  and  B  are  real  positive  quantities  each  less  than  27lfr,  that  is,  less 
than  the  order  of  the  function. 

In  particular,  for  functions  of  the  second  order,  we  have 

z1  +  z,  =  Az  2&>  +  Bz  2&/  +  2.2-0, 
where  Az  and  Bz  are  positive  quantities  each  less  than  2.     Similarly,  if  a  and 

b  be  the  zeros, 

a  +  b  =  Aa  2w  +  £a  2w'  +  2*o, 

where  J.ffl  and  Ba  are  each  less  than  2  ;  hence,  if 

^i  +  ^2  —  a  —  b  —  w2w  +  m'2o>', 

then  w  may  have  any  one  of  the  three  values  - 1,  0,  1  and  so  may  m',  the 
simultaneous  values  not  being  necessarily  the  same. 

Let  a  and  ft  be  the  infinities  of  a  function  of  the  second  class ;  then 
a  +  /3  —  a  —  b  =  ?i2&)  +  n"2w', 

where  n  and  ri  may  each  have  any  one  of  the  three  values  —  1,  0,  1.  By 
changing  the  origin  of  the  fundamental  parallelogram,  so  as  to  obtain  a 
different  set  of  irreducible  points,  we  can  secure  that  n  and  n'  are  zero, 

and  then 

a  +  @  =  a+b. 

Thus,  if  n  be  1   with  an  initial  parallelogram,  so  that 

a  +  /3  =  a  +  &+2&>, 

we  should  take  either  /3  -  2&>  =  {¥,  or  a  -  2&>  =  a',  according  to  the  position  of 
a  and  /3,  and  then  have  a  new  parallelogram  such  that 
a  +  @'  =  a  +  b,  or  a'  +  ft  =  a  +  b. 

The  case  of  exception  is  when  the  function  is  of  the  first  class  and  has  a 
repeated  zero. 

*  The  foregoing  proof  is  suggested  by  Konigsberger,  Theorie  der  elliptischen  Functionen, 
t.  i,  p.  342 ;  other  proofs  are  given  by  Briot  and  Bouquet  and  by  Liouville,  to  whom  the  adopted 
form  of  the  theorem  is  due.  The  theorem  is  substantially  contained  in  one  of  Abel's  general 
theorems  in  the  comparison  of  transcendents. 


116.]  OF   THE   SECOND   ORDER  231 

VIII.  Let  $  (z)  be  a  doubly  -periodic  function  of  the  second  order.  If  7 
be  the  one  double  infinity  when  the  function  is  of  the  first  class,  and  if  a  and  ft 
be  the  two  simple  infinities  when  the  function  is  of  the  second  class,  then  in  the 
former  case 


and  in  the  latter  case  </>  (z)  —  <£  (a  +  (3  —  z). 

Since  the  function  is  of  the  second  order,  so  that  it  has  two  irreducible 
infinities,  there  are  two  (and  only  two)  irreducible  points  in  a  fundamental 
parallelogram  at  which  the  function  can  assume  any  the  same  value  :  let 
them  be  z  and  z'. 

Then,  for  the  first  class  of  functions,  we  have 
z  +  z'  =  27 

=  27  +  2mo>  +  2wa>', 

where  m  and  n  are  integers  ;  and  then,  since  <f)(z)  =  <j>  (z'}  by  definition  of  z 
and  /,  we  have 

<£  (z)  =  <£  (27  -  z  +  2ma) 

=  0(27-4 
For  the  second  class  of  functions,  we  have 

z  +  z  =  a.  +  /3 


so  that,  as  before, 

(/>  (z)  =  </>  (a  +  /3  -  z  +  2ma)  +  2wa>') 


117.  Among  the  functions  which  have  the  same  periodicity  as  a  given 
function  </>  (z),  the  one  which  is  most  closely  related  to  it  is  its  derivative 
<£'  (z).  We  proceed  to  find  the  zeros  and  the  infinities  of  the  derivative  of  a 
function,  in  particular,  of  a  function  of  the  second  order. 

Since  (f>  (z)  is  uniform,  an  irreducible  infinity  of  degree  n  for  </>  (z)  is  an 
irreducible  infinity  of  degree  n  -f  1  for  §'  (z).  Moreover  <£'  (z),  being  uniform, 
has  no  infinity  which  is  not  an  infinity  of  </>  (z)  ;  thus  the  order  of  <£'  (z)  is 
2(?i  +  l)  or  its  order  is  greater  than  that  of  cj>(z)  by  an  integer  which 
represents  the  number  of  distinct  irreducible  infinities  of  <£  (z),  no  account 
being  taken  of  their  degree.  If,  then,  a  function  be  of  order  m,  the  order  of 
its  derivative  is  not  less  than  m  +  1  and  is  not  greater  than  2m. 

Functions  of  the  second  order  either  possess  one  double  infinity  so  that 
within  the  parallelogram  they  take  the  form 


— 
and  then  <j>'  (z)  =  —  -  —  +  %'  (*), 


232  ZEROS   OF  THE   DERIVATIVE  [117. 

that  is,  the  infinity  of  (f>(z)  is  the  single  infinity  of  tf>'  '  (z)  and  it  is  of  the 
third  degree,  so  that  cf>'  (z)  is  of  the  third  order  ;  or  they  possess  two  simple 
infinities,  so  that  within  the  parallelogram  they  take  the  form 


and  then  f  W  =  -  G          -  -  _  +  x'  (,), 


that  is,  each  of  the  simple  infinities  of  <£  (z)  is  an  infinity  for  </>'  (z)  of  the 
second  degree,  so  that  <£'  (z)  is  of  the  fourth  order. 

It  is  of  importance  (as  will  be  seen  presently)  to  know  the  zeros  of 
the  derivative  of  a  function  of  the  second  order. 

For  a  function  of  the  first  class,  let  7  be  the  irreducible  infinity  of  the 
second  degree  ;  then  we  have 


and  therefore  $'(2)  =  —  </>'  (^7  —  z). 

Now  </>'  (z)  is  of  the  third  order,  having  7  for  its  irreducible  infinity  in  the 
third  degree  :  hence  it  has  three  irreducible  zeros. 

In  the  foregoing  equation,  take  z  =  7  :  then 

</>'  (7)  =  -$'  (7), 

shewing  that  7  is  either  a  zero  or  an  infinity.     It  is  known  to  be  the  only 
infinity  of  <£'  (z). 

Next,  take  z  =  7  +  &>  ;  then 

<£'  (7  +  &))  =  —  $'  (7  —  a>) 


=  -  <£'  (7  +  G>), 

shewing  that  7  +  &>  is  either  a  zero  or  an  infinity.     It  is  known  not  to  be  an 
infinity  ;  hence  it  is  a  zero. 

Similarly  7  +  &/  and  7  +  <u  +  &/  are  zeros.  Thus  three  zeros  are  obtained, 
distinct  from  one  another  ;  and  only  three  zeros  are  required  ;  if  they  be  not 
within  the  parallelogram,  we  take  the  irreducible  points  homologous  with 
them.  Hence  : 

IX.  The  three  zeros  of  the  derivative  of  a  function,  doubly  -periodic  in 
2eo  and  2eo'  and  having  7  for  its  double  (and  only)  irreducible  infinity,  are 

7  +  &),     7  +  eo',     7  +  w  +  w  . 

For  a  function  of  the  second  class,  let  a  and  /3  be  the  two  simple 
irreducible  infinities;  then  we  have 


and  therefore  <f>'  (z)=  —  <f>'  (a  +  ft  —  z). 


117.]  OF   A   DOUBLY-PERIODIC   FUNCTION  233 

Now  (j)  (z)  is  of  the  fourth  order,  having  a  and  ft  as  its  irreducible 
infinities  each  in  the  second  degree  ;  hence  it  must  have  four  irreducible 
zeros. 

In  the  foregoing  equation,  take  z  =  \(VL  +  ft)  ;  then 


shewing  that  |  (a  +  /3)  is  either  a  zero  or  an  infinity.     It  is  known  not  to  be 
an  infinity  ;  hence  it  is  a  zero. 

Next,  take  z  =  £  (a  +  (3)  +  w  ;  then 

f(}(«t£)+«}  --+'{*(«+£)-••] 

=  -  <£'  &  (a  +  £)  -  to  +  2&>j 

—.+'{*<«+£)+••}, 

shewing  that  |(a  +  /3)  +  &>  is  either  a  zero  or  an  infinity.     As  before,  it  is 
a  zero. 

Similarly  i  (a  +  /3)  +  &>'  and  i  (a  +  /3)  -f  &>  +  &>'  are  zeros.  Four  zeros  are 
thus  obtained,  distinct  from  one  another;  and  only  four  zeros  are  required. 
Hence  : 

X.  The  four  zeros  of  the  derivative  of  a  function,  doubly-periodic  in  2&> 
and  2o)'  and  having  a  and  /3  for  its  simple  (and  only)  irreducible  infinities,  are 

i(a  +  /3),     i(a  +  /3)  +  a>,     i(a  +  /3)  +  ft>',     |-  (a  +  /3)  +  w  +  a/. 

The  verification  in  each  of  these  two  cases  of  Prop.  VII.,  that  the  sum  of 
the  zeros  of  the  doubly-periodic  function  <£'  (z)  is  congruent  with  the  sum  of 
its  infinities,  is  immediate. 

Lastly,  it  may  be  noted  that,  if  zl  and  z^  be  the  two  irreducible  points  for 
which  a  doubly  -periodic  function  of  the  second  order  assumes  a  given  value, 
then  the  values  of  its  derivative  for  z1  and  for  z%  are  equal  and  opposite.  For 

(j>  (z)  =  <f>  (a  +  /3  -  z)  =  cf>  (z,  +  z.2  -  z), 
since  zl  +  z.,  =  a  +  (3  ;  and  therefore 

<f>  (z)  =  -$'  (z,  +  z.2-  z), 
that  is,  <£'  (zl)  =  —  </>'  (z2), 

which  proves  the  statement. 

118.     We  now  come  to  a  different  class  of  theorems. 

XI.  Any  doubly  -periodic  function  of  the  second  order  can  be  expressed 
algebraically  in  terms  of  an  assigned  doubly-periodic  function  of  the  second 
order,  if  the  periods  be  the  same. 

The  theorem  will  be  sufficiently  illustrated  and  the  line  of  proof 
sufficiently  indicated,  if  we  express  a  function  (/>  (z)  of  the  second  class,  with 
irreducible  infinities  a,  ft  and  irreducible  zeros  a,  b  such  that  a  +  (3  =  a  +  b,  in 


234  FUNCTIONS  [118. 

terms  of  a  function   <£  of  the  first  class  with  7  as  its  irreducible  double 
infinity. 

n     ..       ,     ,. 

Consider  a  function 


Q  (z  +  h)  _ 

A  zero  of  <X>  (z  +  h)  is  neither  a  zero  nor  an  infinity  of  this  function  ;  nor 
is  an  infinity  of  <1>  (z  +  h)  a  zero  or  an  infinity  of  the  function.  It  will  have 
a  and  6  for  its  irreducible  zeros,  if 

a  +  h  =  h', 
b  +  h  +  h'  =  27  ; 

and  these  will  be  the  only  zeros,  for  <E>  is  of  the  second  order.     It  will  have  o 
and  yS  for  its  irreducible  infinities,  if 


and  these  will  be  the  only  infinities,  for  <£  is  of  the  second  order.     These 
equations  are  satisfied  by 


Hence  the  assigned  function,  with  these  values  of  h,  has  the  same  zeros 
and  the  same  infinities  as  $>(z);  and  it  is  doubly-periodic  in  the  same  periods. 
The  ratio  of  the  two  functions  is  therefore  a  constant,  by  Prop.  IV.,  so  that 

c|>  (z  +  h)  —  <I>  (h') 

If  the  expression  be  required  in  terms  of  <&  (z)  alone  and  constants,  then 
<j>  (z  4.  h}  must  be  expressed  in  terms  of  <I>  (z)  and  constants  which  are  values 
of  <X>  (z)  for  special  values  of  z.  This  will  be  effected  later. 

The  preceding  proposition  is  a  special  case  of  a  more  general  theorem 
which  will  be  considered  later ;  the  following  is  another  special  case  of  that 
theorem :  viz. : 

XII.  A  doubly -periodic  function  with  any  number  of  simple  infinities  can 
be  expressed  either  as  a  sum  or  as  a  product,  of  functions  of  the  second  order 
and  the  second  class  which  are  doubly-periodic  in  the  same  periods. 

Let  «j,  «2, ...,  an  be  the  irreducible  infinities  of  the  function  <£,  and 
suppose  that  the  fractional  part  of  <t>  (z)  is 

•A-i       ,       A2      [   ^     i-+     ^n 

z  —  ttj     z  —  «2  z  —  an ' 

with    the    condition    A1  +  A2  + +  An  =  Q.      Let    <j>n(z)    be    a   function, 

doubly-periodic  in  the  same  periods,  with  a,-,  a,-  as  its  only  irreducible  infinities, 


118.]  OF   THE   SECOND   ORDER  235 

supposed   simple;    where  i  and  j   have    the    values    1, ,n.      Then    the 

fractional  parts  of  the  functions  ^>j,  (z),  <£23  (z), . . .  are 

0, 

G, 


i      z  —  a., 

I 
\z  —  «2     ^  —  «, 


respectively;   and  therefore  the  fractional  part  of 

^!^     /    \    , 
•      0»  W  + 


is  Al         An-  An~l 


z-a.!     Z-CL,  z-cin-T.  z-an 

•Ai  An_^         An 

=  -    -+...+-       -  +  — ^, 

Z-Cl!  Z-  «„_!        Z  -  Ctn 


n 

since  S  -4*  =  0.    This  is  the  same  as  the  fractional  part  of  <l>  (z);  and  therefore 


-  <^>23  (f)  -  ...  -  -~ 


has  no  fractional  part.  It  thus  has  no  infinity  within  the  parallelogram  ;  it 
is  a  doubly-periodic  function  and  therefore  has  no  infinity  anywhere  in  the 
plane;  and  it  is  therefore  merely  a  constant,  say  B.  Hence,  changing  the 
constants,  we  have 

$>(z)-B^(z}-B.><t>v(z)-...-Bn-,<t>n-,,n(z}  =  B, 

giving  an  expression  for  <$>  (z}  as  a  linear  combination  of  functions  of  the 
second  order  and  the  second  class.  But  as  the  assignment  of  the  infinities  is 
arbitrary,  the  expression  is  not  unique. 

For  the  expression  in  the  form  of  a  product,  we  may  denote  the  n 
irreducible  zeros,  supposed  simple,  by  «!,...,«„.  We  determine  n  -  2  new 
irreducible  quantities  c,  such  that 


C2= 


Cn—2  —  &n—\  ~r  Cn—3  ~  Q"n—i  > 
Cln  =    ttn    +  C_    —  Q"— 


n 

this  being  possible  because  2  o^  =  2  ar ;  and  we  denote  by  $  (z ;  a,  ft ;  e,  f)  a 

»•=!          r=l 

function  of  .gr,  which  is  doubly-periodic  in  the  periods  of  the  given  function, 


ALGEBRAICAL   RELATIONS  [118. 

has  a  and  $  for  simple  irreducible  infinities  and  has  e  and  /  for  simple 
irreducible  zeros.     Then  the  function 

<f)(z;  al5  «2 ;  «i,  Ci)  <f>  (z ;  as,  ci  ;  0-2,  c2)  ...<£  (2 ;  «n,  cn_2 ;  an_l5  an) 
has  neither  a  zero  nor  an  infinity  at  c1}  at  c2, ...,  and  at  cn_2 ;  it  has  simple 
infinities  at  al}  a2,  ...,  an,  and  simple  zeros  at  alt  a2,  ...,  an-1}  an.  Hence  it 
has  the  same  irreducible  infinities  and  the  same  irreducible  zeros  in  the  same 
degree  as  the  given  function  <£  (z) ;  and  therefore,  by  Prop.  IV.,  <I>  (z)  is 
a  mere  constant  multiple  of  the  foregoing  product. 

The  theorem  is  thus  completely  proved. 

Other  developments  for  functions,  the  infinities  of  which  are  not  simple, 
are  possible ;  but  they  are  relatively  unimportant  in  view  of  a  theorem, 
Prop.  XV.,  about  to  be  proved,  which  expresses  any  periodic  function  in 
terms  of  a  single  function  of  the  second  order  and  its  derivative. 

XIII.  If  two  doubly -periodic  functions  have  the  same  periods,  they  are 
connected  by  an  algebraical  equation. 

Let  u  be  one  of  the  functions,  having  n  irreducible  infinities,  and  v  be 
the  other,  having  m  irreducible  infinities. 

By  Prop.  VI.,  Corollary  I.,  there  are  n  irreducible  values  of  z  for  a  value 
of  u;  and  to  each  irreducible  value  of  z  there  is  a  doubly-infinite  series  of. 
values  of  z  over  the  plane.  The  function  v  has  the  same  value  for  all  the 
points  in  any  one  series,  so  that  a  single  value  of  v  can  be  associated  uniquely 
with  each  of  the  irreducible  values  of  z,  that  is,  there  are  n  values  of  v  for 
each  value  of  u.  Hence,  (§  99),  v  is  a  root  of  an  algebraical  equation  of  the 
nth  degree,  the  coefficients  of  which  are  functions  of  u. 

Similarly  u  is  a  root  of  an  algebraical  equation  of  the  mth  degree,  the 
coefficients  of  which  are  functions  of  v. 

Hence,  combining  these  results,  we  have  an  algebraical  equation  between 
u  and  v  of  the  nth  degree  in  v  and  the  mth  in  u,  where  m  and  n  are  the 
respective  orders  of  v  and  u. 

COROLLARY  I.  If  both  the  functions  be  even  functions  of  z,  then  n  and  m 
are  even  integers ;  and  the  algebraical  relation  between  u  and  v  is  of  degree  ^n 
in  v  and  of  degree  ^m  in  u. 

COROLLARY  II.  If  a  function  u  be  doubly-periodic  in  &>  and  &>',  and  a 
function  v  be  doubly -periodic  in  fl  and  U',  where 

n  =  mca  +  nta,  I!'  =  m'w  +  nw! , 
m,  n,  m',  n  being  integers,  then  there  is  an  algebraic  relation  between  u  and  v. 

119.  It  has  been  proved  that,  if  a  doubly-periodic  function  u  be  of  order  m, 
then  its  derivative  du/dz  is  doubly-periodic  in  the  same  periods  and  is  of  an 
order  n,  which  is  not  less  than  m  +  1  and  not  greater  than  2?/i.  Hence,  by 


119.]  BETWEEN   HOMOPERIODIC   FUNCTIONS  237 

Prop.  XIII.,  there  subsists  between  u  and  u  an  algebraical  equation  of  order  m 
in  u'  and  of  order  n  in  u;  let  it  be  arranged  in  powers  of  u'  so  that  it  takes 
the  form 

U"  u'm  _j_  JJ  u'm—i  _i      _  _    i     U   _  u'2    i    JJ    _u'   i     JJ     __  Q 

where  U0,  U1}  ... ,  Um  are  rational  integral  algebraical  functions  of  u  one  at 
least  of  which  must  be  of  degree  n. 

Because  the  only  distinct  infinities  of  u'  are  infinities  of  u,  it  is  impossible 
that  u'  should  become  infinite  for  finite  values  of  u:  hence  U0  =  0  can  have  no 
finite  roots  for  u,  that  is,  it  is  a  constant  and  so  it  may  be  taken  as  unity. 

And  because  the  m  values  of  z,  for  which  u  assumes  a  given  value,  have 
their  sum  constant  save  as  to  integral  multiples  of  the  periods,  we  have 

corresponding  to  a  variation  8u ;  or 

du      du  du 

f/7/ 

Now    —  is  one  of  the  values  of  u'  corresponding  to  the  value  of  u,  and  so  for 

the  others  ;  hence 

3    1 


r=i  ur 
that  is,  by  the  foregoing  equation, 

"  m— i 


=  0, 


un. 

and  therefore  Um-^  vanishes.     Hence : 

XIV.     There  is  a  relation,  between  a  doubly -periodic  function  u  of  order  m 
and  its  derivative,  of  the  form 

u'm  +  U^'™-1  +  ...+  U^u'*  +  Um  =  0, 

where  Ul}...,  Um_2,  Um  are  rational  integral  algebraical  functions  of  u,  at 
least  one  of  which  must  be  of  degree  n,  the  order  of  the  derivative,  and  n  is 
not  less  than  m  +  1  and  not  greater  than  2m. 

Further,  by  taking  v  =  -  ,  which  is  a  function  of  order  m  because  it  has  the 

Uj 

m  irreducible  zeros  of  u  for  its  infinities,  and  substituting,  we  have 

vf™  _  03  U^'m~l  +  v*U«v'm~*  -  . . .  ±  v2"1-4  Um_2v''2  +  v2"1  Um  =  0. 
The  coefficients  of  this  equation  must  be  integral  functions  of  v ;  hence  the 
degree  of  Ur  in  u  cannot  be  greater  than  2r. 

COROLLARY.     The  foregoing  equation  becomes  very  simple  in  the  case  of 
doubly-periodic  functions  of  the  second  order. 

Then  m  =  2. 


238  DIFFERENTIAL   EQUATION  [119. 

If  the  function  have  one  infinity  of  the  second  degree,  its  derivative  has 
that  infinity  in  the  third  degree,  and  is  of  the  third  order,  so  that  n  =  3  ;  and 
the  equation  is 

/y7?/\2 

(  ^  )  =  \u?  +  3/iw2  +  Svu  +  p, 
\d*J 

where  X,  /*,  v,  p  are  constants.     If  6  be  the  infinity,  so  that 

A 
*.£(,)_-_—  +  £(*), 

where  %  (^)  is  everywhere  finite  in  the  parallelogram,  then  -  =  ±A  ;  and  the 

/77/ 

zeros  of  -j-  are  6  +  o>,  0  +  &/,  6  +  o>  +  CD'  ;  so  that 
diz 

a,')}  { 


This  is  £/ie  general  differential  equation  of  Weierstrasss  elliptic  functions. 

If  the  function  have  two  simple  infinities  a  and  @,  its  derivative  has  each 
of  them  as  an  infinity  of  the  second  degree,  and  is  of  the  fourth  order,  so  that 
n  =  4  ;  and  the  equation  is 


(du\*  _ 
(dz)  = 


dM  +    c2w  +  >c3u  +  c4, 
where  c0,  c1}  c2,  c3,  c4  are  constants.     Moreover 


where  ^  (^)  is  finite  everywhere  in  the  parallelogram.     Then  cu  =  G~2  ;  and 

^/'i/ 

the  zeros  of  -y-  are  ^  (a  +  /3),  -|-  (a  +  (3)  +  w,  ^  (a  -f  /3)  +  cof,  %  (a  +  ft)  +  w  +  &>', 
ft/2 

so  that  the  equation  is 


(«  +  13)+  «  +  «}]. 
This  is  the  general  differential  equation  of  Jacobis  elliptic  functions. 

The  canonical  forms  of  both  of  these  equations  will  be  obtained  in  Chapter 
XI.,  where  some  properties  of  the  functions  are  investigated  as  special  illustra 
tions  of  the  general  theorems. 

Note.  All  the  derivatives  of  a  doubly-periodic  function  are  doubly- 
periodic  in  the  same  periods,  and  have  the  same  infinities  as  the  function  but 
in  different  degrees.  In  the  case  of  a  function  of  the  second  order,  which 
must  satisfy  one  or  other  of  the  two  foregoing  equations,  it  is  easy  to  see  that 
a  derivative  of  even  rank  is  a  rational,  integral,  algebraical  function  of  u,  and 
that  a  derivative  of  odd  rank  is  the  product  of  a  rational,  integral,  algebraical 
function  of  u  by  the  first  derivative  of  u. 


119.]  OF   DOUBLY-PERIODIC   FUNCTIONS  239 

It  may  be  remarked  that  the  form  of  these  equations  confirms  the  result 
at  the  end  of  §  117,  by  giving  two  values  of  u'  for  one  value  of  u,  the  two 
values  being  equal  and  opposite. 

Ex.     If  u  be  a  doubly-periodic  function  having  a  single  irreducible  infinity  of  the  third 
degree  so  as  to  be  expressible  in  the  form 

2      6 
—  -o  +  -5  +  integral  function  of  z 

z       z 

within  the  parallelogram  of  periods,  then  the  differential  equation  of  the  first  order  which 
determines  u  is 


where  £74  is  a  quartic  function  of  u  and  where  a  is  a  constant  which  does  not  vanish  with  6. 

(Math.  Trip.,  Part  II,  1889.) 

XV.  Every  doubly  -periodic  function  can  be  expressed  rationally  in  terms 
of  a  function  of  the  second  order,  doubly-periodic  in  the  same  periods,  and  its 
derivative. 

Let  u  be  a  function  of  the  second  order  and  the  second  class,  having  the 
same  two  periods  as  v,  a  function  of  the  rath  order  ;  then,  by  Prop.  XIII., 
there  is  an  algebraical  relation  between  u  and  v  which,  being  of  the  second 
degree  in  v  and  the  mth  degree  in  u,  may  be  taken  in  the  form 

Lv*  -  2Mv  +  P  =  0, 

where  the  quantities  L,  M,  P  are  rational,  integral,  algebraical  functions  of  u 
and  at  least  one  of  them  is  of  degree  m.     Taking 

Lv-M=w, 

we  have  w2  =  M'2  —  LP, 

a  rational,  integral,  algebraical  function  of  u  of  degree  not  higher  than  2w. 

Thus  w  cannot  be  infinite  for  any  finite  value  of  u  :  an  infinite  value  of  u 
makes  w  infinite,  of  finite  multiplicity.  To  each  value  of  u  there  correspond 
two  values  of  w  equal  to  one  another  but  opposite  in  sign. 

Moreover  w,  being  equal  to  Lv  -  M  ,  is  a  uniform  function  of  z,  say  F(z\ 
while  it  is  a  two-valued  function  of  u.  A  value  of  u  gives  two  distinct 
values  of  z,  say  zl  and  £2  ;  hence  the  values  of  w,  which  arise  from  an  assigned 
value  of  u,  are  values  of  w  arising  as  uniform  functions  of  the  two  distinct 
values  of  z.  Hence  as  the  two  values  of  w  are  equal  in  magnitude  and 
opposite  in  sign,  we  have 

r(4)+J*(4)-Oi 

that  is,  since  ^  +  z.2  =  a.  +  ft  where  a  and  /3  are  the  irreducible  infinities  of  u, 


so  that  l(a  +  £),  £(a  +  /3)  +  a>,  £(a  +  £)  +  «',  and  £  (a  +  /3)  +  a>  +  a>'  are  either 
zeros  or  infinities  of  w.  They  are  known  not  to  be  infinities  of  u,  and  w  is 
infinite  only  for  infinite  values  of  u  ;  hence  the  four  points  are  zeros  of  w. 


240  RELATIONS   BETWEEN  [119. 

But  these  are  all  the  irreducible  zeros  of  u' ;  hence  the  zeros  of  u'  are 
included  among  the  zeros  of  w. 

Now  consider  the  function  w/u'.  The  numerator  has  two  values  equal 
and  opposite  for  an  assigned  value  of  u ;  so  also  has  the  denominator.  Hence 
w/u'  is  a  uniform  function  of  u. 

This  uniform  function  of  u  may  become  infinite  for 
(i)  infinities  of  the  numerator, 
(ii)  zeros  of  the  denominator. 

But,  so  far  as  concerns  (ii),  we  know  that  the  four  irreducible  zeros  of  the 
denominator  are  all  simple  zeros  of  u'  and  each  of  them  is  a  zero  of  w .;  hence 
w/u'  does  not  become  infinite  for  any  of  the  points  in  (ii).  And,  so  far  as 
concerns  (i),  we  know  that  all  of  them  are  infinities  of  u.  Hence  w/u,  a 
uniform  function  of  u,  can  become  infinite  only  for  an  infinite  value  of  u,  and 
its  multiplicity  for  such  a  value  is  finite;  hence  it  is  a  rational,  integral, 
algebraical  function  of  u,  say  N,  so  that 

w  =  Nu'. 

Moreover,  because  w2  is  of  degree  in  u  not  higher  than  2m,  and  u'2  is  of 
the  fourth  degree  in  u,  it  follows  that  N  is  of  degree  not  higher  than  m  —  2. 

We  thus  have  Lv  —  M  —  Nu, 

M+Nu      M     N  , 

v=  ~r  =  L  +  LU> 

where  L,  M,  N  are  rational,  integral,  algebraical  functions  of  u ;  the  degrees 
of  L  and  M  are  not  higher  than  m,  and  that  of  N  is  not  higher  than  m  —  2. 

Note  1.  The  function  u,  which  has  been  considered  in  the  preceding 
proof,  is  of  the  second  order  and  the  second  class.  If  a  function  u  of  the 
second  order  and  the  first  class,  having  a  double  irreducible  infinity,  be 
chosen,  the  course  of  proof  is  similar ;  the  function  w  has  the  three  irreducible 
zeros  of  u'  among  its  zeros  and  the  result,  as  before,  is 

w  =  Nu'. 

But,  now,  w"-  is  of  degree  in  u  not  higher  than  2m  and  u'2  is  of  the  third 
degree  in  u  ;  hence  N  is  of  degree  not  higher  than  m  —  2  and  the  degree  of  w2 
in  u  cannot  be  higher  than  2m  —  1. 

Hence,  if  L,  M,  P  be  all  of  degree  m,  the  terms  of  degree  2m  in  LP  —  M2 
disappear.  If  all  of  them  be  not  of  degree  m,  the  degree  of  M  must  not  be 
higher  than  m  —  l  ;  the  degree  of  either  L  or  P  must  be  m,  but  the  degree 
of  the  other  must  not  be  greater  than  m—l,  for  otherwise  the  algebraical 
equation  between  u  and  v  would  not  be  of  degree  m  in  u. 

We  thus  have 

Lv2  -  2Mv  +  P  =  (),    Lv  -  M  =  Nu', 


119.]  HOMOPERIODIC   FUNCTIONS  241 

where  the  degree  of  N  in  u  is  not  higher  than  m  —  2.  If  the  degree  of  L  be 
less  than  TO,  the  degree  of  M  is  not  higher  than  TO  —  1  and  the  degree  of  P  is 
TO.  If  the  degree  of  L  be  m,  the  degree  of  M  may  also  be  m  provided  that  the 
degree  of  P  be  TO  and  that  the  highest  terms  be  such  that  the  coefficient 
of  u2m  in  LP  -  M'2  vanishes. 

Note  2.  The  theorem  expresses  a  function  v  rationally  in  terms  of  u  and 
u  :  but  u'  is  an  irrational  function  of  u,  so  that  v  is  not  expressed  rationally 
in  terms  of  u  alone. 

But,  in  Propositions  XI.  and  XII.,  it  was  indicated  that  a  function  such  as 
v  could  be  rationally  expressed  in  terms  of  a  doubly-periodic  function,  such  as 
u.  The  apparent  contradiction  is  explained  by  the  fact  that,  in  the  earlier 
propositions,  the  arguments  of  the  function  u  in  the  rational  expression  and 
of  the  function  v  are  not  the  same  ;  whereas,  in  the  later  proposition  whereby 
v  is  expressed  in  general  irrationally  in  terms  of  u,  the  arguments  are  the 
same.  The  transition  from  the  first  (which  is  the  less  useful  form)  to  the 
second  is  made  by  expressing  the  functions  of  those  different  arguments  in 
terms  of  functions  of  the  same  argument  when  (as  will  appear  subsequently,  in 
§  121,  in  proving  the  so-called  addition-theorem)  the  irrational  function  of  u, 
represented  by  the  derivative  u,  is  introduced. 

COROLLARY  I.  Let  H  denote  the  sum  of  the  irreducible  infinities  or  of 
the  irreducible  zeros  of  the  function  u  of  the  second  order,  so  that  H  =  2y  for 
functions  of  the  first  class,  and  O  =  a  +  /3  for  functions  of  the  second  class. 
Let  u  be  represented  by  <f>  (z)  and  v  by  ty  (z),  when  the  argument  must  be  put 
in  evidence.  Then 


so  that  W-Z)  = 

J-j  i_j      ±j 

Hence  ^  (z)  +  ^  (fl  -  z)  =  2  ^=  2R, 

JL 


First,  if  y  (z)  =  ,Jr  (ft  -  z\  then  S  =  0  and  ^  (z)  =  R  :  that  is,  a  function  ^  (z), 
which  satisfies  the  equation 


can  be  expressed  as  a  rational  algebraical  meromorphic  function  of  <f>  (z)  of  the 
second  order,  doubly  -periodic  in  the  same  periods  and  having  the  sum  of  its 
irreducible  infinities  congruent  with  O. 

Second,  if  ^  (e)  =  -  y,  (fl  _  z\  then  R  =  0  and  ^  (*)  =  flf  (*)  ;  that  is, 
function  ^  (z),  which  satisfies  the  equation 


16 


a 


242  HOMOPERIODIC   FUNCTIONS  [119. 

can  be  expressed  as  a  rational  algebraical  meromorphic  function  of  <£  (z), 
multiplied  by  0'  (z),  where  $  (z}  is  doubly-periodic  in  the  same  periods,  is  of  the 
second  order,  and  has  the  sum  of  its  irreducible  infinities  congruent  with  Q. 

Third,  if  ty(z)  have  no  infinities  except  those  of  u,  it  cannot  become 
infinite  for  finite  values  of  u  ;  hence  L  =  0  has  no  roots,  that  is,  L  is  a  constant 
which  may  be  taken  to  be  unity.  Then  i/r  (z)  a  function  of  order  m  can  be 
expressed  in  the  form 


where,  if  the  function  </>  (z)  be  of  the  second  class,  the  degree  of  M  is  not 
higher  than  m  ;  but,  if  it  be  of  the  first  class,  the  degree  of  M  is  not  higher 
than  m  -  1  ;  and  in  each  case  the  degree  of  N  is  not  higher  than  m  -  2. 

It  will  be  found  in  practice,  with  functions  of  the  first  class,  that  these 
upper  limits  for  degrees  can  be  considerably  reduced  by  counting  the  degrees 
of  the  infinities  in 


Thus,  if  the  degree  of  M  in  u  be  ^  and  of  N  be  \  the  highest  degree  of  an  : 
infinity  is  either  2/t  or  2X  +  3  ;  so  that,  if  the  order  of  ^  (z)  be  m,  we  should 

have 

m  =  2/j,  or  m  =  2\  +  3,     > 

according  as  m  is  even  or  odd. 

When  functions  of  the  second  class  are  used  to  represent  a  function  ^r  (z), 
which  has  two  infinities  a  and  /3  each  of  degree  n,  then  it  is  easy  to  see  that 
M  is  of  degree  n  and  N  of  degree  n  -  2  ;  and  so  for  other  cases. 

COROLLARY  II.  Any  doubly  -periodic  function  can  be  expressed  rationally 
in  terms  of  any  other  function  u  of  any  order  n,  doubly-periodic  in  the  same 
periods,  and  of  its  derivative  ;  and  this  rational  expression  can  always  be  taken 
in  the  form 

U0  +  U,U'  +  t/X3  +  •  •  •  +  Un-,u'n~\ 

where  U0,  ...  ,  £7n-i  are  algebraical,  rational,  meromorphic  functions  of  u. 

COROLLARY  III.  If  <f)  be  a  doubly-periodic  function,  then  <f>  (u  +  v)  can  be 
expressed  in  the  form 


where  ^  is  a  doubly  -periodic  function  in  the  same  periods  and  of  the  second 
order  :  each  of  the  functions  A,  D,  E  is  a  symmetric  function  of^(u)  and  i/r  (v), 
and  B  is  the  same  function  of^(v)  and  ty(u)  as  C  is  of  ty  (u)  and  ty  (v). 

The  degrees  of  A  and  E  are  not  greater  than  m  in  ty  (u)  and  than  m  in  ^  (v), 
where  m  is  the  order  of  </>  ;  the  degree  of  D  is  not  greater  than  m  -  2  in  ^  (u) 
and  than  m  -  2  in  ^  (v)  ;  the  degree  of  B  is  not  greater  than  m  -  2  in  ^  (u) 
and  than  m  in  ^  (v),  and  the  degree  of  C  is  not  greater  than  m  -  2  in  -^  (v) 
and  than  m  in  -^  (u). 


CHAPTER  XI. 

DOUBLY-PERIODIC  FUNCTIONS  OF  THE  SECOND  ORDER. 

THE  present  chapter  will  be  devoted,  in  illustration  of  the  preceding 
theorems,  to  the  establishment  of  some  of  the  fundamental  formulae  relating 
to  doubly-periodic  functions  of  the  second  order  which,  as  has  already  (in 
§  119,  Cor.  to  Prop.  XIV.)  been  indicated,  are  substantially  elliptic  functions  : 
but  for  any  development  of  their  properties,  recourse  must  be  had  to  treatises 
on  elliptic  functions. 

It  may  be  remarked  that,  in  dealing  with  doubly-periodic  functions,  we 
may  restrict  ourselves  to  a  discussion  of  even  functions  and  of  odd  functions. 
For,  if  (/>  (z)  be  any  function,  then  £  {<j>  (z}  +  <j>(—  z}}  is  an  even  function,  and 
\  {(f>(z)  —  </>(—  z}}  is  an  odd  function,  both  of  them  being  doubly-periodic  in 
the  periods  of  <f>  (z)  ;  and  the  new  functions  would,  in  general,  be  of  order 
double  that  of  <J>(z).  We  shall  practically  limit  the  discussion  to  even 
functions  and  odd  functions  of  the  second  order. 

120.  Consider  a  function  <j>(z\  doubly-periodic  in  2&>  and  2w';  and  let 
it  be  an  odd  function  of  the  second  class,  with  a  and  ft  as  its  irreducible 
infinities,  and  a  and  b  as  its  irreducible  zeros*. 

Then  we  have  <£  (z)  =  (f>  (a  +  /3  —  z) 

which  always  holds,  and  <f>  (—  z)  =  —  </>  (z) 

which  holds  because  <£  (z)  is  an  odd  function.     Hence 

<f>  (a  +  /3  +  z)  =  (/>(-  *) 

=  -$(*) 

so  that  a  +  ft  is  not  a  period  ;  and 


-*(*), 

To  fix  the  ideas,  it  will  be  convenient  to  compare  it  with  snz,  for  which  2w  =  4^T,  2<a'  = 
a=iK',  p=iK'  +  2K,  a-0,  and  b  =  2K. 

16—2 


244  DOUBLY-PERIODIC   FUNCTIONS  [120. 

whence  2  (a  +  /S)  is  a  period.     Since  a  -f  /3  is  not  a  period,  we  take  a  +  /3  =  a>, 
or  =  &)',  or  =  &>  +  w'  ;  the  first  two  alternatives  merely  interchange  &>  and  &>',  so 

that  we  have  either 

a  +  /3  =  o), 

or  a  +  /3  =  ft)  +  &>'. 

And  we  know  that,  in  general, 

a  +  b  =  a  +  /3. 
First,  for  the  zeros  :  we  have 


so  that  </>(0)  is  either  zero  or  infinite.     The  choice  is  at  our  disposal;  for 

-  satisfies  all  the  equations  which  have  been  satisfied  by  $(z)  and  an 

</>(*) 

infinity  of  either  is  a  zero  of  the  other.     We  therefore  take 


so  that  we  have  a  =  0, 

6  =  to    or    &)  +  ft)'. 
Next,  for  the  infinities  :  we  have 

*(*)—$(-*) 

and  therefore  <j>  (-  a)  =  -  $  (a)  =  oo  . 

The  only  infinities  of  <£  are  a  and  /3,  so  that  either 

—  a=  a, 

or  -CL  =  P. 

The  latter  cannot  hold,  because  it  would  give  a  +  /3  =  0  whereas 

or  =  &>  +  &/;  hence 

2a  =  0, 

which  must  be  associated  with  a  +  /3  =  w  or  with  a  +  /3  =  &>  +  &/. 

Hence  a,  being  a  point  inside  the  fundamental  parallelogram,  is  either  0, 
a),  &)',  or  tw  +  &)'. 

It  cannot  be  0  in  any  case,  for  that  is  a  zero. 

If  a  _|_  ^  =  Wj  then  a  cannot  be  tw,  because  that  value  would  give  ft  =  0, 
which  is  a  zero,  not  an  infinity.  Hence  either  a  =  «',  and  then  /3  =  &/  +  &>; 
or  a  =  &)'  +  &),  and  then  /3  =  ft)'.  These  are  effectively  one  solution  ;  so  that,  if 

a  +  /3  =  &),  we  have 

a,  /3  =  ft)',  &>'  +  &)) 

and  a,  6  =  0,  &)          )  ' 

jf  a  +  /S  =  w  +  &>',  then  a  cannot  be  CD  f  &)',  because  that  value  would  give 
{$  =  0,  which  is  a  zero,  not  an  infinity.  Hence  either  a  =  &>  and  then  ft  =  &)', 
or  a  =  ft)'  and  then  /3  =  &).  These  again  are  effectively  one  solution  ;  so  that, 

if  a  +  /3  =  &)  +  &>',  we  have 

a,  £  =  o),  ft)' 
and  a,  6  =  0,  «o  +  ft)') 


120.]  OF  THE   SECOND   CLASS  245 

This  combination  can,  by  a  change  of  fundamental  parallelogram,  be  made 
the  same  as  the  former  ;  for,  taking  as  new  periods 

2ft/  =  2a>'t         2fl  =  2«  +  2a>', 
which  give  a  new  fundamental  parallelogram,  we  have  a  +  j3  =  H,  and 

a,  ft  —  &>',  ft  —  ft/,  that  is,  ft/,  ft  —  03'  +  2<o' 
so  that  a,  /3  =  ft/,  O  +  a/] 

and  a,  b  =  0, 

being  the  same  as  the  former  with  O  instead  of  &>.     Hence  it  is  sufficient  to 
retain  the  first  solution  alone  :  and  therefore 

a  =  to',         ft  =  CD'  +  co, 
a  =  0,  6  =  w. 

Hence,  by  §  116,  1.,  we  have 


where  F(z)  is  finite  everywhere  within  the  parallelogram. 

Again,  $ (z  +  a/)  has  z  =  0  and  z  =  &>  as  its  irreducible  infinities,  and 
it  has  2  =  0)'  and  z  =  &>  +  &/  as  its  irreducible  zeros,  within  the  parallelogram 
of  (f)  (z}  ;  hence 


where  ^  (2)  is  finite  everywhere  within  the  parallelogram.     Thus 


a  function  which  is  finite  everywhere  within  the  parallelogram  ;  since  it  is 
doubly-periodic,  it  is  finite  everywhere  in  the  plane  and  it  is  therefore  a 
constant  and  equal  to  the  value  at  any  point.  Taking  -  i&/  as  the  point 
(which  is  neither  a  zero  nor  an  infinity)  and  remembering  that  </>  is  an  odd 
function,  we  have 

*  (*)*(*  +  «0  =  -  ft  (*»')}'  =  p 

k  being  a  constant  used  to  represent  the  value  of  -  {<£  (^o/)}"2. 

Also  <j>(z  +  o))  =  <f>(z  +  a  +  /3-  2&/) 

=  c/>0  +  a  +  /3)=-(£  (z), 
and  therefore  also  <£  (&>  —  z)  =  <f)  (z). 

The  irreducible  zeros  of  <j>'  (z)  were  obtained  in  §  117,  X.  In  the 
present  example,  those  points  are  a>'  +  £ft>,  &>'  +  ffc>,  £&>,  f  &>  ;  so  that,  as 
there,  we  have 

£to'('W-{*(i)-4>(i*yito(*)-HW 

where  K  is  a  constant.     But 

$  (f®)  =  0  (2®  -  lft>)  =  (f)  (-!«)  =  _(£  (1  a,)  ; 


246  DOUBLY-PERIODIC   FUNCTIONS  [120. 

and  0(fw  +  &/)  =  <£(2a>  +  2w'  -.!&>-  a/) 

=  <£(-  2  <o  -to') 

=  —  </>(£&>  +  &>'); 

so  that    •  ,.  .  4  I  - 


where  J.  is  a  new  constant,  evidently  equal  to  {<£'(0)}2.  Now,  as  we  know 
the  periods,  the  irreducible  zeros  and  the  irreducible  infinities  of  the  function 
</>  (z),  it  is  completely  determinate  save  as  to  a  constant  factor.  To  determine 
this  factor  we  need  only  know  the  value  of  <$>(z)  for  any  particular  finite 
value  of  z.  Let  the  factor  be  determined  by  the  condition 


then,  since  <£(^ft>)<£(^G>  +  ft/)  =  T 

by  a  preceding  equation,  we  have 


and  then 

ft'  (*)}»  -  {f  (0)}«  [1  -  {<£  (*)}2]  [1  -  fr  {(/>  (*)}'] 


Hence,  since  (/>  (2)  is  an  odd  function,  we  have 

<£  (z)  =  sn  (//,£). 

Evidently  2/xtu,  2/^ft)'  =  4^T,  2^',  where  K  and  ^T'  have  the  ordinary  signifi 
cations.     The  simplest  case  arises  when  /A  =  1. 

121.  Before  proceeding  to  the  deduction  of  the  properties  of  even 
functions  of  z  which  are  doubly-periodic,  it  is  desirable  to  obtain  the 
addition-theorem  for  <f>,  that  is,  the  expression  of  <p  (y  +  z)  in  terms  of 
functions  of  y  alone  and  z  alone. 

When  <f>  (y  +  z)  is  regarded  as  a  function  of  z,  which  is  necessarily  of  the 
second  order,  it  is  (§  119,  XV.)  of  the  form 


where  M  and  L  are  of  degree  in  <£  (z)  not  higher  than  2  and  N  is  independent 
of  z.  Moreover  y  +  z  =  a  and  y  +  z  =  ft  are  the  irreducible  simple  infinities 
of  <j)  (y  +  z)  ;  so  that  L,  as  a  function  of  z,  may  be  expressed  in  the  form 


and  therefore 

Z±_^(iHL^^^)}l 
(z)  - 


121.]  OF  THE  SECOND   CLASS  247 

where  P,  Q,  R,  S  are  independent  of  z  but  they  may  be  functions  of  y.     Now 

</>  (a  -  y)  =  </>  (w'  -  y)  =  - 


and  <£  (/3  —  y)  =  <j>  (&>'  +  w  —  y}  = 


so  that  the  denominator  of  the  expression  for  <f>  (y  +  2)  is 


Since  </>  (z)  is  an  odd  function,  <£'  (#)  is  even  ;  hence 

,A   p  - 
~  */ 


and  therefore       $  (y  +  z)  —  $  (y  —  z)  =  -  - 


Differentiating  with  regard  to  z  and  then  making  z  =  0,  we  have 


so  that,  substituting  for  Q  we  have 


Interchanging  y  and  z  and  noting  that  </>  (t/  —  z)  =  —  (f)  (z  —  y),  we  have 


md  therefore  d>  C7y  *  Z}  d>'  (0}  - 

W+*)1>< 


which  is  the  addition-theorem  required. 

Ex.  If  f(u)  be  a  doubly-periodic  function  of  the  second  order  with  infinities  61}  i2, 
and  0(tt)  a  doubly  -periodic  function  of  the  second  order  with  infinities  alt  a2  such  that, 
in  the  vicinity  of  «»  (for  i  —  1,  2),  we  have 

^  (M)  =  ,7~!r  +Pi+&  (u~ai)  +  ......  > 

c6  —  u-j 

thon  /M-/W  =  •  i»  W+*  W-ft-ftl- 

the  periods  being  the  same  for  both  functions.     Verify  the  theorem  when  the  functions  are 
sn  u  and  sn  (u  +  v}.  (Math.  Trip.  Part  II.,  1  891.) 

Prove  also  that,  for  the  function  $  (u),  the  coefficients  p±  and  p2  are  equal.     (Burnside.) 

122.     The    preceding   discussion   of    uneven    doubly-periodic   functions 
having  two  simple  irreducible  infinities  is  a  sufficient  illustration  of  the 


248  DOUBLY-PERIODIC   FUNCTIONS  [122. 

method  of  procedure.  That,  which  now  follows,  relates  to  doubly- periodic 
functions  with  one  irreducible  infinity  of  the  second  degree ;  and  it  will  be 
used  to  deduce  some  of  the  leading  properties  of  Weierstrass's  er-function 
(of  §  57)  and  of  functions  which  arise  from  it. 

The  definition  of  the  <r-function  is 


where  fi  =  2ma>  +  2m'a)',  the  ratio  of  &>'  :  &>  not  being  purely  real,  and  the 
infinite  product  is  extended  over  all  terms  that  are  given  by  assigning  to 
m  and  to  m'  all  positive  and  negative  integral  values  from  +00  to  —  oo , 
excepting  only  simultaneous  zero  values.  It  has  been  proved  (and  it  is 
easy  to  verify  quite  independently)  'that,  when  cr(z)  is  regarded  as  the 
product  of  the  primary  factors 


the  doubly-infinite  product  converges  uniformly  and  unconditionally  for  all 
values  of  z  in  the  finite  part  of  the  plane ;  therefore  the  function  which  it 
represents  can,  in  the  vicinity  of  any  point  c  in  the  plane,  be  expanded  in  a 
converging  series  of  positive  powers  of  z  —  c,  but  the  series  will  only  express 
the  function  in  the  domain  of  c.  The  series,  however,  can  be  continued  over 
the  whole  plane. 

It  is  at  once  evident  that  a-  (z)  is  not  a  doubly-periodic  function,  for  it  has 
no  infinity  in  any  finite  part  of  the  plane. 

It  is  also  evident  that  a  (z)  is  an  odd  function.  For  a  change  of  sign  in  z 
in  a  primary  factor  only  interchanges  that  factor  with  the  one  which  has 
equal  and  opposite  values  of  m  and  of  m',  so  that  the  product  of  the  two  factors 
is  unaltered.  Hence  the  product  of  all  the  primary  factors,  being  independent 
of  the  nature  of  the  infinite  limits,  is  an  even  function ;  when  z  is  associated 
as  a  factor,  the  function  becomes  uneven  and  it  is  a-  (z). 

The  first  derivative,  a'  (z),  is  therefore  an  even  function ;  and  it  is  not 
infinite  for  any  point  in  the  finite  part  of  the  plane. 

It  will  appear  that,  though  a-  (z)  is  not  periodic,  it  is  connected  with 
functions  that  have  2o>  and  2&>'  for  periods ;  and  therefore  the  plane  will  be 
divided  up  into  parallelograms.  When  the  whole  plane  is  divided  up,  as  in 
§  105,  into  parallelograms,  the  adjacent  sides  of  which  are  vectorial  repre 
sentations  of  2w  and  2&/,  the  function  a-(z)  has  one,  and  only  one,  zero  in 
each  parallelogram;  each  such  zero  is  simple,  and  their  aggregate  is  given 
by  z  =  £l.  The  parallelogram  of  reference  can  be  chosen  so  that  a  zero 
of  <r  (z}  does  not  lie  upon  its  boundary ;  and,  except  where  explicit  account  is 


122.]  OF   THE   FIRST   CLASS  249 

taken  of  the  alternative,  we  shall  assume  that  the  argument  of  &>'  is  greater 
than  the  argument  of  to,  so  that  the  real  part*  of  w'/ia)  is  positive. 

123.  We  now  proceed  to  obtain  other  expressions  for  a-  (z),  and  particu 
larly,  in  the  knowledge  that  it  can  be  represented  by  a  converging  series  in 
the  vicinity  of  any  point,  to  obtain  a  useful  expression  in  the  form  of  a  series, 
converging  in  the  vicinity  of  the  origin. 

Since  er  (z)  is  represented  by  an  infinite  product  that  converges  uniformly 
and  unconditionally  for  all  finite  values  of  z,  its  logarithm  is  equal  to  the  sum 
of  the  logarithms  of  its  factors,  so  that 


where  the  series  on  the  right-hand  side  extends  to  the  same  combinations  of 
m  and  m'  as  the  infinite  product  for  z,  and,  when  it  is  regarded  as  a  sum  of 

z          z^  (         z\ 

functions  o  +  i  7^2  +  ^°£  (  ^  ~  r>  ) »  ^ne  sei>ies  converges  uniformly  and  uncon- 

__  --  \  1  - , 

ditionally,  except  for  points  z  =  £l.     This  expression  is  valid  for  log  a  (z)  over 
the  whole  plane. 

Now  let  these  additive  functions  be  expanded,  as  in  §  82.     In  the  imme 
diate  vicinity  of  the  origin,  we  have 


a  series  which  converges  uniformly  and    unconditionally    in    that  vicinity. 
Then  the  double  series  in  the  expression  for  log  a  (z}  becomes 


and  as  this  new  series  converges  uniformly  and  unconditionally  for  points  in 
the  vicinity  of  z  =  0,  we  can,  as  in  §  82,  take  it  in  the  form 

oo       ~r    (    oo        oo  } 

5"       J  5*    y  O-n 

—   4    —  \  ^->     <5r  »•      () 

r=3  r    (-00  -oo  J 

which  will  also,  for  such  values  of  z,  converge  uniformly  and  unconditionally. 
In  §  56,  it  was  proved  that  each  of  the  coefficients 

00  00 

2  s  n-*-, 


—  00     -  00 


for  r  =  3,  4,...,  is  finite,  and  has  a  value  independent  of  the  nature  of  the 
infinite  limits  in  the  summation.  When  we  make  the  positive  infinite  limit 
for  m  numerically  equal  to  the  negative  infinite  limit  for  m,  and  likewise  for 


This  quantity  is  often  denoted  by  ffi  (  .  -  J . 


250  WEIERSTRASS'S  [123. 

ra',  then  each  of  these  coefficients  determined  by  an  odd  index  r  vanishes, 
and  therefore  it  vanishes  in  general.     We  then  have 

log  a-  0)  =  log  z  -  I*  22ft-4  -  ^  22ft-6  -  ^  22ft-8 

a  series  which  converges  uniformly  and  unconditionally  in  the  vicinity  of  the 
origin. 

The  coefficients,  which  occur,  involve  «o  and  «',  two  independent  constants. 
It  is  convenient  to  introduce  two  other  magnitudes,  g.2  and  g3,  denned  by  the 

equations 

#2=  6022ft-4,     #3  =  140220-0, 

so  that  g2  and  </3  are  evidently  independent  of  one  another;  then  all  the 
remaining  coefficients  are  functions*  of  g.2  and  g3.     We  thus  have 


and  therefore  <r  (z)  =  ze   m 

where  the  series  in  the  index,  containing  only  even  powers  of  z,  converges 

uniformly  and  unconditionally  in  the  vicinity  of  the  origin. 

It  is  sufficiently  evident  that  this  expression  for  a- (z)  is  an  effective 
representation  only  in  the  vicinity  of  the  origin ;  for  points  in  the  vicinity  of 
any  other  zero  of  cr  (z),  say  c,  a  similar  expression  in  powers  of  z  -  c  instead 
of  in  powers  of  z  would  be  obtained. 

124.     From  the  first  form  of  the  expression  for  log  cr  (z),  we  have 


o-(z)      z      _«,  _ 

where  the  quantity  in  the  bracket  on  the  right-hand  side  is  to  be  regarded  as 
an  element  of  summation,  being  derived  from  the  primary  factor  in  the 
product-expression  for  cr  (z\ 


We  write  £(z)  =     ,  ^  , 

so  that  %(z)  is,  by  §  122,  an  odd  function,  a  result  also  easily  derived  from  the 
foregoing  equation ;  and  so 


This  expression  for  £  (z)  is  valid  over  the  whole  plane. 
Evidently  £  (z)  has  simple  infinities  given  by 

for  all  values  of  ra  and  of  m  between  +  oo  and  -  oo  ,  including  simultaneous 
zeros.  There  is  only  one  infinity  in  each  parallelogram,  and  it  is  simple ;  for 
the  function  is  the  logarithmic  derivative  of  a  (z\  which  has  no  infinity  and 

*  See  Quart.  Journ.,  vol.  xxii.,  pp.   4,  5.     The  magnitudes  g2  and  g3  are  often  called  the 
invariants. 


124.]  ELLIPTIC   FUNCTION  251 

only  one  zero  (a  simple  zero)  in  the  parallelogram.     Hence  %(z)  is  not  a 
doubly-periodic  function. 

For  points,  which  are  in  the  immediate  vicinity  of  the  origin,  we  have 


but,  as  in  the  case  of  cr(z),  this  is  an  effective  representation  of  %(z)  only 
in  the  vicinity  of  the  origin  ;  and  a  different  expression  would  be  used  for 
points  in  the  vicinity  of  any  other  infinity. 

We  again  introduce  a  new  function  g>  (z)  defined  by  the  equation 


Because  £  is  an  odd  function,  $  (z)  is  an  even  function  ;  and 


where  the  quantity  in  the  bracket  is  to  be  regarded  as  an  element  of 
summation.  This  expression  for  $  (z)  is  valid  over  the  whole  plane. 
Evidently  |p  (z)  has  infinities,  each  of  the  second  degree,  given  by  z  =  fl, 
for  all  values  of  m  and  of  m  between  -f  oo  and  -  oo  ,  including  simultaneous 
zeros  ;  and  there  is  one,  and  only  one,  of  these  infinities  in  each  parallelogram. 
One  of  these  infinities  is  the  origin;  using  the  expression  which  represents 
log  a-  (z)  in  the  immediate  vicinity  of  the  origin,  we  have 


=  -2  +  20  9**  +  ^  9*?+  •  •  • 

for   points   z   in   the   immediate  vicinity  of  the   origin.     A   corresponding 
expression  exists  for  g>  (z)  in  the  vicinity  of  any  other  infinity. 

125.     The   importance   of  the  function   $  (z)   is   due    to  the  following 
theorem  :  — 

The  function  $>  (z)  is  doubly-periodic,  the  periods  being  2<w  and  2&/. 
Wo  have  -l 


where  the  doubly-infinite  summation  excludes  simultaneous  zero  values,  and 
the  expression  is  valid  over  the  whole  plane.     Hence 

+  ^-n  -  Si 


252  WEIERSTRASS'S  [125. 

so  that 


obtained  by  combining  together  the  elements  of  the  summation  in  g>  (z  +  2<w) 
and  |p  (,z).  The  two  terms,  not  included  in  the  summation,  can  be  included, 
if  we  remove  the  numerical  restriction  as  to  non-admittance  of  simultaneous 
zero  values  for  m  and  m'\  and  then 

2.)  -  f  (,)  =  2 


_ 


where  now  the  summation  is  for  all  values  of  m  and  of  m'  from  +  oo  to  —  oo  . 
Let  q  denote  the  infinite  limit  of  m,  and  p  that  of  m'.  Then  terms  in  the 
first  fraction,  for  0  =  2  (mm  +  m'w'},  are  the  same  as  terms  in  the  second  for 
£1  =  2  (m  —  l)w  +  2m'  w  ;  cancelling  these,  we  have 

m'=p 

-fC    = 


where  q  is  infinite.     But 


?r)2      sin2  c ' 
and  therefore 

»»'  =  p  =  oo  1  ^.2 

2i 


-  2mV}2      W    . 
sin 


2o/ 

if/)  be  infinitely  great  compared  with  q.  This  condition  may  be  assumed  for 
the  present  purpose,  because  the  value  of  g>  (z)  is  independent  of  the  nature 
of  the  infinite  limits  in  the  summation  and  is  therefore  unaffected  by  such  a 
limitation. 

f         -  ""  1       ]    f  £+!?(9+1)       -*$-*  * 

l_*  J  '       l_ 

The  fraction  —,  has  a  real  part.     In  the  exponent  it  is  multiplied  by  q  +  1. 

that  is,  by  an  infinite  quantity ;  so  that  the  real  part  of  the  index  of 
the  exponential  is  infinite,  either  positive  or  negative.  Thus  either  the 
first  term  is  infinite  and  the  second  zero,  or  vice  versa;  in  either  case, 

r  T  i  • 

sin    \z  +  2  (q  +  1)  twl  ^— ,     is  infinite,  and  therefore 
2o)  J 


{2  +  2(q  +  l)(o-  2m  V}2 
Similarly  for  the  other  sum.     Hence 


=  0. 


In  the  same  way  it  may  be  shewn  that 

£>0  +  2a/)-£>0)  =  0; 

therefore  £>  (z)  is  doubly-periodic  in  2<o  and  2a>'. 


126.]  ELLIPTIC   FUNCTION  253 

Now  in  any  parallelogram  whose  adjacent  sides  are  2&>  and  2&>',  there  is 
only  one  infinity  and  it  is  of  multiplicity  two;  hence,  by  §  116,  Prop.  III., 
Cor.  3,  2o)  and  2&>'  determine  a  primitive  parallelogram  for  $>  (z). 

We  shall  assume  the  parallelogram  of  reference  chosen  so  as  to  include 
the  origin. 

126.     The  function  $  (z)  is  thus  of  the  second  order  and  the  first  class. 

Since  its  irreducible  infinity  is  of  the  second  degree,  the  only  irreducible 
infinity  of  g>'  (z}  is  of  the  third  degree,  being  the  origin  ;  and  the  function 
<§t  '  (z)  is  odd. 

The  zeros  of  jp'  (z}  are  thus  &>,  ft/,  and  (&>  +  to')  ;  or,  if  we  introduce  a  new 
quantity  w"  defined  by  the  equation 

&>"  =  &)  +  &>', 

the  zeros  of  <@!  (z)  are  &>,  &>',  &>". 
We  take 

#>(«)  =  e1}        p(a>")  =  e2,        p(m')  =  e3,        %>(z)  =  p: 
and  then,  by  §  119,  Prop.  XIV.,  Cor.,  we  have 


where  A  is  some  constant.     To  determine  the  equation  more  exactly,  we 
substitute  the  expression  of  jp  in  the  vicinity  of  the  origin.     Then 


80  that  P'  =  -j+iQff*  + 

When  substitution  is  made,  it  is  necessary  to  retain  in  the  expansion  all 
terms  up  to  z°  inclusive.     We  then  have,  for  |p'2,  the  expression 

4      2          4 


and  for  A  (^  -  e-,)  (p  -  e2)  (p  -  e3),  the  expression 

A  r1   3  9-  3 

1L^6  +  20^+285r3+>" 

-  (e,  +  e2  +  e3)(^+—g2  +  ...)+  (6le,  +  e2e3  +  e&)  (-  +  ...)-  tf,«A  | 

When  we  equate  coefficients  in  these  two  expressions,  we  find 


e1  +  e2  +  e3  =  0,         e&  +  e» 
therefore  the  differential  equation  satisfied  by  p  is 


254  PERIODICITY  [126. 

Evidently  £>"  =  6§>2  -  %gs, 


and  so  on  ;  and  it  is  easy  to  verify  that  the  2wth  derivative  of  g)  is  a  rational 
integral  algebraical  function  of  <p  of  degree  n  +  l  and  that  the  (2w+l)th 
derivative  of  fp  is  the  product  of  g>'  by  a  rational  integral  algebraical  function 
of  degree  n. 

The  differential  equation  can  be  otherwise  obtained,  by  dependence  on 
Cor.  2,  Prop.  V.  of  §  116.     We  have,  by  differentiation  of  %>', 


for  points  in  the  vicinity  of  the  origin  ;  and  also 

^+!^2  +  r4^2  +  "-- 

Hence  <@"  and  §>2  have  the  same  irreducible  infinities  in  the  same  degree  and 
their  fractional  parts  are  essentially  the  same  :  they  are  homoperiodic  and 
therefore  they  are  equivalent  to  one  another.  It  is  easy  to  see  that  g>"  —  6(jp2 
is  equal  to  a  function  which,  being  finite  in  the  vicinity  of  the  origin,  is  finite 
in  the  parallelogram  of  reference  and  therefore,  as  it  is  doubly  -periodic,  is 
finite  over  the  whole  plane.  It  therefore  has  a  constant  value,  which  can  be 
obtained  by  taking  the  value  at  any  point;  the  value  of  the  function  for 

z  =  0  is  —  \g»  and  therefore 

g>"_6^  =  -^2, 

so  that  |p"=  6g»2-|<72, 

the  integration  of  which,  with  determination  of  the  constant  of  integration, 

leads  to  the  former  equation. 

This  form,  involving  the  second  derivative,  is  a  convenient  one  by  which 
to  determine  a  few  more  terms  of  the  expansion  in  the  vicinity  of  the  origin  : 
and  it  is  easy  to  shew  that 


from  which  some  theorems  relating  to  the  sums  ^SH"2*1  can  be  deduced*. 

Ex.     If  cn  be  the  coefficient  of  22n~2  in  the  expansion  of  $  (z)  in  the  vicinity  of  the 
origin,  then 

c»=/o~  .  iw..     ON    2    Crfin-r-  (Weierstrass.) 


We  have  jp'2  =  4^>3  -  g$  -  g3 ; 

the  function  jjp'  is  odd  and  in  the  vicinity  of  the  origin  we  have 


*  See  a  paper  by  the  author,  Quart.  Journ.,  vol.  xxii,  (1887),  pp.  1 — 43,  where  other  references 
are  given  and  other  applications  of  the  general  theorems  are  made. 


126.]  OF   WEIERSTRASS'S   FUNCTION  255 

hence,  representing  by  —  (4|p3  —  g$>—  g^  that  branch  of  the  function  which  is 
negative  for  large  real  values,  we  have 


and  therefore  z  = 

The  upper  limit  is  determined  by  the  fact  that  when  z  =  0,  g>  =  oo  ;  so  that 

-  r     d® 

_  r  d%> 

lp  {4  (p  -  ej  (p  -  e2)  (p  -  e,)}*' 

This  is,  as  it  should  be,  an  integral  with  a  doubly-infinite  series  of  values. 
We  have,  by  Ex.  6  of  §  104, 

r 

0)j  =  ft)       =| 

J<h 


, 

ft>3=    ft)       = 

J 

with  the  relation  a)"  =  a)  +  co'. 

127.     We  have  seen  that  g>  (z)  is  doubly-periodic,  so  that 

p(*+2»)  =  $>(*), 

and  therefore  dg(5  + 2«)  =  dgW 

a^  dz 

hence  integrating  ?(^  +  2<»)  =  %(z)  +  A. 

Now  ^  is  an  odd  function ;  hence,  taking  z  —  —  co  which  is  not  an  infinity  of  £, 
we  have 

^  =  2^(&))=27; 

say,  where  r)  denotes  £  (&>) ;  and  therefore 

£(*-*•  2»)r- £•(*)«  89, 

which  is  a  constant. 

Similarly  %(z  +  2&>')  -  ^  (^)  -  2i/, 

where  r;'  =  ^ (to')  and  is  constant. 

Hence  combining  the  results,  we  have 

%  (z  +  2w&)  +  2?rc  V)  -£(z)  =  2mri  +  Zmrj', 
where  m  and  m  are  any  integers. 

It  is  evident  that  77  and  rj'  cannot  be  absorbed  into  £;  so  that  £  is  not  a 
periodic  function,  a  result  confirmatory  of  the  statement  in  §  124. 


256  PSEUDO-PERIODIC  [127. 

There  is,  however,  a  pseudo-periodicity  of  the  function  £ :  its  characteristic 
is  the  reproduction  of  the  function  with  an  added  constant  for  an  added 
period.  This  form  is  only  one  of  several  simple  forms  of  pseudo-periodicity 
which  will  be  considered  in  the  next  chapter. 

128.  But,  though  %(z)  is  not  periodic,  functions  which  are  periodic  can 
be  constructed  by  its  means. 

Thus,  if      4>(z)=AS(z-a)+Bt(z-V) 
then  *  +  2w-(*)  =  2A£(*-a 


so  that,  subject  to  the  condition 

A+B+C+...=0, 
<j)  (z)  is  a  doubly-periodic  function. 

Again,  we  know  that,  within  the  fundamental  parallelograTH,  f  has  a 
single  irreducible  infinity  and  that  the  infinity  is  simple;  hence  the  irre 
ducible  infinities  of  the  function  </>  (V)  are  z  =  a,  b,  c,  ...,  and  each  is  a  simple 
infinity.  The  condition  A  +  B  +  C  +  ...=0  is  merely  the  condition  of  Prop. 
III.,  §  116,  that  the  '  integral  residue '  of  the  function  is  zero. 

Conversely,  a  doubly-periodic  function  with  m  assigned  infinities  can  be 
expressed  in  terms  of  f  and  its  derivatives.  Let  ax  be  an  irreducible  infinity 
of  <£>  of  degree  n,  and  suppose  that  the  fractional  part  of  <I>  for  expansion  in 
the  immediate  vicinity  of  ax  is 

A!  i?i        |        ^       KI 

Then 

-if  (*- 4).-... 


is  not  infinite  for  z  =  a^. 

Proceeding  similarly  for  each  of  the  irreducible  infinities,  we  have  a 
function 


r 

which  is  not  infinite  for  any  of  the  points  z  =  alt  a2,  ....     But  because  <f>  (z) 
is  doubly-periodic,  we  have 

and  therefore  the  function 


128.] 


FUNCTIONS 


257 


is  doubly-periodic.  Moreover,  all  the  derivatives  of  any  order  of  each  of  the 
functions  £  are  doubly-periodic;  hence  the  foregoing  function  is  doubly- 
periodic. 

The  function  has  been  shewn  to  be  not  infinite  at  the  points  a1}  a2,  ..., 
and  therefore  it  has  no  infinities  in  the  fundamental  parallelogram  ;  con 
sequently,  being  doubly-periodic,  it  has  no  infinities  in  the  plane  and  it  is 
a  constant,  say  G.  Hence  we  have 


g, 


r=i 


r=i 


m 

with  the  condition    2  Ar  =  0,  which  is   satisfied  because  <E>  (z)  is  doubly- 
periodic. 

This  is  the  required  expression  *  for  <I>  (z)  in  terms  of  the  function  %  and 
its  derivatives;  it  is  evidently  of  especial  importance  when  the  indefinite 
integral  of  a  doubly-periodic  function  is  required. 

129.  Constants  77  and  77',  connected  with  &>  and  «',  have  been  introduced 
by  the  pseudo-periodicity  of  £(z)\  the  relation,  contained  in  the  following 
proposition,  is  necessary  and  useful  :  — 

The  constants  77,  w',  &>,  &>'  are  connected  by  the  relation 


the  +  or  -  sign  being  taken  according  as  the  real  part  of  o>'fa)i  is  positive  or 
negative. 

A  fundamental  parallelogram  having  an  angular  point  at  z0  is  either  of 

the  form  (i)  in  fig.  34,  in   which  case  9t  f-^]  is 

\mj 

positive  :  or  of  the  form  (ii),  in  which  case  9J  ( — . ) 

\Ct)l/ 

is  negative.  Evidently  a  description  of  the  paral 
lelogram  A  BCD  in  (i)  will  give  for  an  integral  the 
'same  result  (but  with  an  opposite  sign)  as  a  de 
scription  of  the  parallelogram  in  (ii)  for  the  same 
integral  in  the  direction  A  BCD  in  that  figure. 

We  choose  the  fundamental  parallelogram,  so 
that  it  may  contain  the  origin  in  the  included 
area.  The  origin  is  the  only  infinity  of  £  which 
can  be  within  the  area :  along  the  boundary  £  is 
always  finite. 

Now  since 

*  See  Hermitet  Ann.  de  Toulouse,  t.  ii,  (1888),  C,  pp.  1—12. 
F. 


20+2o)' 


Fig.   34 


258  PSEUDO-PERIODICITY   OF   WEIERSTRASS's  [129. 

the  integral  of  £(z)  round  ABCD  in  (i),  fig.  34,  is  (§  116,  Prop.  II.,  Cor.) 

rD  CB 

2r)dz  -        fy'dz, 

J  A  J  A 

the  integrals  being  along  the  lines  AD  and  AB  respectively,  that  is,  the 

integral  is 

4  (rju>'  —  rfw}. 

But  as  the  origin  is  the  only  infinity  within  the  parallelogram,  the  path  of 
integration  ABCD  A  can  be  deformed  so  as  to  be  merely  a  small  curve  round  ! 
the  origin.  In  the  vicinity  of  the  origin,  we  have 


and  therefore,  as  the  integrals  of  all  terms  except  the  first  vanish  when  taken 
round  this  curve,  we  have 


=  2-Trt. 

Hence  4  (rjw  —  TJ'O))  =  27ri, 

and  therefore  i](f>  —  rju>  =  \iri. 

This  is  the  result  as  derived  from  (i),  fig.  34,  that  is,  when  91  [?-)  is  positive. 

\i/tU/ 

When  (ii),  fig.  34,  is  taken  account  of,  the  result  is  the  same  except 
that,  when  the  circuit  passes  from  z0  to  z0  +  2&>,  then  to  z0  +  2t»  +  2o>', 
then  to  z0  +  2&>'  and  then  to  z0,  it  passes  in  the  negative  direction  round  the' 
parallelogram.  The  value  of  the  integral  along  the  path  ABCDA  is  the 
same  as  before,  viz.,  4  (rjw  —  rj'a))  ;  when  the  path  is  deformed  into  a  small! 

rdz 

curve  round  the  origin,  the  value  of  the  integral  is  I  —  taken  negatively,  an 

J     ** 

therefore  it  is  —  2?ri  :  hence 

t](£)  —  rj  (a  =  —  \Tri. 

Combining  the  results,  we  have 

rjay'  —  f]w  =  ±  ^Tri, 
/    '\ 
according:  as  9t  (  —  .  ]  is  positive  or  negative. 

\0)lJ 

COROLLARY.  If  there  be  a  change  to  any  other  fundamental  parallelo 
gram,  determined  by  2H  and  2O',  where 

£1  =  pa)  +  qa)',          £1'  =  p'co  +  q'a)', 
p,  q,  p',  q'  being  integers  such  that  pq  —  p'q  =  ±  1,  and  if  H,  H'  denote 

C(ft'),  then 

H  =  pr}  +  qrj',          H'  =  p'rj  +  q'f}'  ; 

therefore  HW  -  H'£l  =  ±  \-iri, 


according  as  the  real  part  of  T^  is  positive  or  negative. 


130.]  PRODUCT-FUNCTIONS  259 

130.     It  has  been  seen  that  £  (z)  is  pseudo-periodic  ;  there  is  also  a  pseudo- 
periodicity  for  o-  (z),  but  of  a  different  kind.     We  have 


that  is, 

0-0  + 

and  therefore  a- (z  +  2<w)  =  Ae^zcr  (z), 

where  A  is  a  constant.     To  determine  A,  we  make  z  =  —  &>,  which  is  not  a 
zero  or  an  infinity  of  a  (z)  ;  then,  since  a  (z)  is  an  odd  function,  we  have 

so  that  o-  0  +  2&>)  =  -  e*>  <z+<0>  a-  (z). 

Hence  o-(z  +  4eo)  =  —  e*> (z+3ft))  <r(z  +  2&>) 


and  similarly  a  (z  +  2mey)  =  (—  l)m 

Proceeding  in  the  same  way  from 


we  find  a~(z  +  2m  V)  =  (-  l)m'  e*>'  <w/z+m'2w')  o-  (z). 

Then         a  (z  +  2ma>  +  2m  V)  =  (-  l)m  e2^  (ww+^»+»»»»V)  Q-  (^  +  2m/eo/) 

==  /  _  J  \m+m'  g  sz  (mij+m'V)  +2'?m2<o+47]mmV+27)'m'2co'  _. 


But  lyct)'  —  r/o)  =  ±  \iri, 

SO  that  g2mm'(r|a>'—  rj'w)  _  e±mm'iri  _  /_  |\nj.m' 

and  therefore 


2m  V)  =  (-  l)w 

which  is  the  law  of  change  of  a  (z)  for  increase  of  z  by  integral  multiples  of 
the  periods. 

Evidently  <r(z)  is  not  a  periodic  function,  a  result  confirmatory  of  the 
statement  in  §  122.  But  there  is  a  pseudo-periodicity  the  characteristic  of 
which  is  the  reproduction,  for  an  added  period,  of  the  function  with  an 
exponential  factor  the  index  being  linear  in  the  variable.  This  is  another 
of  the  forms  of  pseudo-periodicity  which  will  be  considered  in  the  next 
chapter. 

131.  But  though  <r(z)  is  not  periodic,  we  can  by  its  means  construct 
functions  which  are  periodic  in  the  pseudo-periods  of  a  (z). 

By  the  result  in  the  last  section,  we  have 

<r  (z  —  a.  +  2ma>  +  2m'&)')      cr  (z  —  a)          +  ,, 
<r(z-fi  +  2mo>  +  2m V)  ~  <r(z  ^~J3)  &  ' 

17—2 


260  DOUBLY-PERIODIC    FUNCTIONS  [131. 

and  therefore,  if  <f>  (z)  denote 

a-  (z  —  cti)  a  (z  —  02) cr  (z  —  &») 


then  $  (z  +  2ra&>  +  2m  V)  =  e2(m>}  +m'*'>  <2^-2^ 

so  that  $  (z)  is  doubly-periodic  in  2«  and  2&>'  provided 


Now  the  zeros  of  <f>(z),  regarded  as  a  product  of  o--functions,  are  als  a2,...,  «„ 
and  the  points  homologous  with  them  ;  and  the  infinities  are  Pi,  /32, ... ,  ftn  and  i 
the  points  homologous  with  them.     It  may  happen  that  the  points  a  and  ft  j 
are    not    all    in    the    parallelogram   of  reference ;    if  the    irreducible   points 
homologous  with  them  be  a1}  ...,  an  and  blt ... ,  bn,  then 

Sar  =  ~S.br  (mod.  2&>,  2co'), 

and  the  new  points  are  the  irreducible  zeros  and  the  irreducible  infinities  of 
<}>(z).     This  result,  we  know  from  Prop.  III.,  §  116,  must  be  satisfied. 

It  is  naturally  assumed  that  no  one  of  the  points  a  is  the  same  as,  or  is 
homologous  with,  any  one  of  the  points  ft :  the  order  of  the  doubly-periodic 
function  would  otherwise  be  diminished  by  1. 

If  any  a  be  repeated,  then  that  point  is  a  repeated  zero  of  <j>(z);  similarly- 
if  any  ft  be  repeated,  then  that  point  is  a  repeated  infinity  of  <£  (z).    In  every, 
case,  the  sum  of  the  irreducible  zeros  must  be  congruent  with  the  sum  of  the 
irreducible  infinities  in  order  that  the  above  expression  for  <j)(z)  may  be 
doubly-periodic. 

Conversely,  if  a  doubly-periodic  function  <£  (z)  be  required  with  m  assignedJ 
irreducible  zeros  a  and  m  assigned  irreducible  infinities  b,  which  are  subject t 

to  the  congruence 

2a  =  26  (mod.  2co,  2&>'), 

we  first  find  points  OL  and  ft  homologous  with  a  and  with  b  respectively  sucht 
that 


rru          +U      t          « 

Then  the  function 


a-  (z  -  Pi) a(z  —  ftm) 

has  the  same  zeros  and  the  same  infinities  as  </>  (z),  and  is  homoperiodic  withi 
it ;  and  therefore,  by  §  116,  IV., 

o-(s-ai) o-^-otm) 

9  \z>  —  •"•  „(„  _  o  \         *(*—ft  V 


where  A  is  a  quantity  independent  of  z. 

Ex.  1.  Consider  ft?  (z).  It  has  the  origin  for  an  infinity  of  the  third  degree  and  all  thti 
remaining  infinities  are  reducible  to  the  origin  ;  and  its  three  irreducible  zeros  are  a,  a/,  a>"  j 
Moreover,  since  <o"=a>'  +  a>,  we  have  w  +  w'  +  w"  congruent  with  but  not  equal  to  zerw 
We  therefore  choose  other  points  so  that  the  sum  of  the  zeros  may  be  actually  the  same, 


131.]  EXAMPLES  261 

as  the  sum  of  the  infinities,  which  is  zero  ;  the  simplest  choice  is  to  take  <»,  &>',  -  «". 
Hence 


where  A  is  a  constant.  To  determine  A,  consider  the  expansions  in  the  immediate 
vicinity  of  the  origin  ;  then 

2  o-  (  -  co)  <r  (  -  to  )  v  (a)") 

?"•"  ......  S3  *  ......  > 

sothat  y^-g^^/rf^'tf. 

O-  («)  or  (eo  )  o-  (a  )  O"3  (2) 

Another  method  of  arranging  zeros,  so  that  their  sum  is  equal  to  that  of  the  infinities, 
is  to  take  —  w,  —  «',  co"  ;  and  then  we  should  find 

dy  M  =2 
r  W 

This  result  can,  however,  be  deduced  from  the  preceding  form  merely  by  changing  the 
sign  of  z. 

Ex.  2.     Consider  the  function 

.  a-  (u  +  v)  a-  (u  —  v) 

*«(«) 

where  v  is  any  quantity  and  A  is  independent  of  u.  It  is,  qua  function  of  u,  doubly- 
periodic  ;  and  it  has  u  =  0  as  an  infinity  of  the  second  degree,  all  the  infinities  being 
homologous  with  the  origin.  Hence  the  function  is  homoperiodic  with  g>  (u)  and  it  has 
the  same  infinities  as  $>  (u)  :  thus  the  two  are  equivalent,  so  that 


where  B  and    C  are  independent  of  u.     The  left-hand   side  vanishes   if  n—v;   hence 
(v),  and  therefore 


where   A'   is   a  new   quantity   independent   of   u.     To  determine   .4'   we   consider   the 
expansions  in  the  vicinity  of  u  =  0  ;  we  have 

A.'<r(v)<r(-v) 


sothat 

and  therefore  cr-     = 

o-2  (%)  o-2  (v) 

a  formula  of  very  great  importance. 

Ex.  3.     Taking  logarithmic  derivatives  with  regard  to  u  of  the  two  sides  of  the  last 
equation,  we  have 


and,  similarly,  taking  them  with  regard  to  D,  we  have 


whence 


262  EXAMPLES  [131. 

giving  the  special  value  of  the  left-hand  side  as  (§  128)  a  doubly-periodic  function.     It  is 
also  the  addition-theorem,  so  far  as  there  is  an  addition-theorem,  for  the  ^-function. 

Ex.  4.    We  can,  by  differentiation,  at  once  deduce  the  addition-theorem  for  g)  (u  +  v). 
Evidently 


which  is  only  one  of  many  forms  :  one  of  the  most  useful  is 


which  can  be  deduced  from  the  preceding  form. 

The  result  can  be  used  to  modify  the  expression  for  a  general  doubly-periodic  function 
*  (z)  obtained  in  §  128.     We  have 


Each  derivative  of  f  can  be  expressed  either  as  an  integral  algebraical  function  of  $  (z  -  a,.) 
or  as  the  product  of  jjp'  (z  —  ar)  by  such  a  function  ;  and  by  the  use  of  the  addition-theorem 
these  can  be  expressed  in  the  form 


L          > 
where  L,  M,  N  are  rational  integral  algebraical  functions  of  $(z).    Hence  the  function 


can  be  expressed  in  the  same  form,  the  simplest  case  being  when  all  its  infinities  are 
simple,  and  then 

4.  (z)  =  C+  2  Ar{(e-ar) 


(*)-§»  (Or) 


with  the  condition  2  Ar  =  0. 

r=l 

Ex.  5.  The  function  $  (z)  —  e1  is  an  even  function,  doubly-periodic  in  2«  and  2o>  and 
having  2  =  0  for  an  infinity  of  the  second  degree  ;  it  has  only  a  single  infinity  of  the  second 
degi'ee  in  a  fundamental  parallelogram. 

Again,  z  =  &>  is  a  zero  of  the  function ;  and,  since  ^X  («)  =  0  but  $>"  (o>)  is  not  zero,  it  is  a 
double  zero  of  $  (z)-el.  All  the  zeros  are  therefore  reducible  to  2  =  o> ;  and  the  function 
has  only  a  single  zero  of  the  second  degree  in  a  fundamental  parallelogram. 

Taking  then  the  parallelogram  of  reference  so  as  to  include  the  points  0  =  0  and  0=o>, 
we  have 


where  Q  (z)  has  no  zero  and  no  infinity  for  points  within  the  parallelogram. 

Again,  for  g>  (z  +  o>)  -  e± ,  the  irreducible  zero  of  the  second  degree  within  the  parallelo- 


131.]  OF   DOUBLY-PERIODIC    FUNCTIONS  263 


gram  is  given  by  S  +  <B  =  O>,  that  is,  it  is  0  =  0;  and  the  irreducible  infinity  of  the  second 
degree  within  the  parallelogram  is  given  by  z  +  a  =  0,  that  is,  it  is  z  =  v.     Hence  we  have 


where  Ql  (z)  has  no  zero  and  no  infinity  for  points  within  the  parallelogram. 

Hence  {£>  (z}  -  ej  {%>  (z  +  «)  -  ej  m  Q  (z)  Q1  (z\ 

that  is,  it  is  a  function  which  has  no  zero  and  no  infinity  for  points  within  the 
parallelogram  of  reference.  Being  doubly-periodic,  it  therefore  has  no  zero  and  no  infinity 
anywhere  in  the  plane  ;  it  consequently  is  a  constant,  which  is  the  value  for  any  point. 
Taking  the  special  value  s  =  a>,  we  have  jp(m')  =  es,  and  (jf>(a>'  +  a>)  =  e2  ;  and  therefore 

{#>  (*)  ~  e,}  (V  (*  +  «)-  e,}  =  (e3  -  *i)  (e,  -  *i). 

Similarly  {#>  (z)  -  e2}  {#>  (z  +  »")  -  e2}  =  (ex  -  e2)  (es  -  e2), 

and  {#>  (2)  -  ^  {p  (z  +  a)')  -  e3}  =  (e2  -  e3}  (^  -  <?3). 

It  is  possible  to  derive  at  once  from  these  equations  the  values  of  the  ^-function  for 
the  quarter-periods. 

Note.  In  the  preceding  chapter  some  theorems  were  given  which  indicated  that 
functions,  which  are  doubly-periodic  in  the  same  periods,  can  be  expressed  in  terms  of  one 
another  :  in  particular  cases,  care  has  occasionally  to  be  exercised  to  be  certain  that  the 
periods  of  the  functions  are  the  same,  especially  when  transformations  of  the  variables  are 
effected.  For  instance,  since  g)  (z)  has  the  origin  for  an  infinity  and  sn  u  has  it  for  a  zero, 
it  is  natural  to  express  the  one  in  terms  of  the  other.  Now  $  (z)  is  an  even  function,  and 
sn  u  is  an  odd  function  ;  hence  the  relation  to  be  obtained  will  be  expected  to  be  one 
between  ®(z)  and  sn2w.  But  one  of  the  periods  of  sn2  u  is  only  one-half  of  the  correspond 
ing  period  of  sn  u  ;  and  so  the  period-parallelogram  is  changed.  The  actual  relation*  is 

(P  (z)  -  <?3  =  (<?!-  e3)  sn-2«, 
where  u  =  (el  -esf  z  and  F  =  (<?2  -e^)l(el  -e:i). 

Again,  with  the  ordinary  notation  of  Jacobian  elliptic  functions,  the  periods  of  sn  z  are 
4  A"  and  2iA",  those  of  dn  z  are  2  A  and  4i  K',  and  those  of  en  z  are  4  A'  and  2A'+  2iA''.  The 
squares  of  these  three  functions  are  homoperiodic  in  2K  and  ZiK'  ;  they  are  each  of  the 
second  order,  and  they  have  the  same  infinities.  Hence  sn2  z,  en2  z,  dn2  z  are  equivalent  to 
one  another  (§  116,  V.). 

But  such  cases  belong  to  the  detailed  development  of  the  theory  of  particular  classes  of 
functions,  rather  than  to  what  are  merely  illustrations  of  the  general  propositions. 

132.  As  a  last  illustration  giving  properties  of  the  functions  just 
considered,  the  derivatives  of  an  elliptic  function  with  regard  to  the  periods 
will  be  obtained. 

Let  (/>  (z)  be  any  function,  doubly-periodic  in  2o>  and  2&/  so  that 

</>  (z  +  2m&>  +  2m  V)  =  </>  (z), 

the  coefficients  in  <f>  implicitly  involve  &>  and  CD'.     Let  <f>1}  <£2,  and  </>'  respec 
tively  denote  90/3t»,  9<£/9o/,  9^/9^  ;  then 

^  (z  +  2771&)  +  2m'  to')  +  %m<j>'  (z  +  2mo>  +  2wV)  =  fa  (z), 
fa  (z  +  2m&)  +  2??iV)  -1-  2m'<£'  (z  +  2rao>  +  2wV)  =  fa  (z), 

$  (z  +  2m&)  +  2m  V)  =  <f>  (z). 

"  Halphen,  Fonctions  Elliptiques,  t.  i,  pp.  23  —  25. 


264  PERIOD-DERIVATIVES  [132. 

Multiplying  by  &>,  ro',  z  respectively  and  adding,  we  have 
&></>!  (z  +  2mo>  +  2m  V)  +  o>'</>2  (z  +  2m<«  +  2mV) 

+  (z  +  2mw  +  2wV)  </>'  0  +  2mo) 

=  0)0!  (Z)  +  0)'(f).2  (Z)  +  Z<f>'  (Z). 

Hence,  if  f(z}  =  mfa  (z)  +  6/02  (z)  +  z$  (z), 

then  f(z)  is  a  function  doubly  -periodic  in  the  periods  of  (f>. 

Again,  multiplying  by  rj,  77',  %(z),  adding,  and  remembering  that 
£0  +  2mm  +  2raV)  =  £($) 


we  have 

77$!  (z  +  2mw  +  2m  V)  +  7/<£2  0  +  2m«o  +  2m  V) 

+  £(z  +  2mm  +  2m'  ay')  <f>'  (z  +  2mw  +  2m'a>') 

-ik<fi)+J+,(*)  +  f(*Wto 

Hence,  if  g  (z)  =  yfa  (z)  +  q'fa  (z)  +  £  (z)  $'  (z), 

then  g(z)  is  a  function  doubly-periodic  in  the  periods  of  <f>. 

In  what  precedes,  the  function  <f>(z)  is  any  function,  doubly-periodic  in 
2o>,  2&)'  ;  one  simple  and  useful  case  occurs  when  0  (z)  is  taken  to  be  the 
function  z.  Now 


and  fW-J-^ 

hence,  in  the  vicinity  of  the  origin,  we  have 

9P        >d@        d&         2 

(o  ^-  +  03  5*7  +  jp-«-  as  —  +  even  integral  powers  of  z1 
d(o          d(o         dz          z- 

=  -2^>, 

since  both  functions  are  doubly-periodic  and  the  terms  independent  of  z 
vanish  for  both  functions.  It  is  easy  to  see  that  this  equation  merely 
expresses  the  fact  that  <p,  which  is  equal  to 

l 


is  homogeneous  of  degree  —  2  in  z,  &>,  to'. 
Similarly 

9|J>       /  ty  ^d%>          22 

77  2+*)  ~r-/  +  b  (-2r)  y-  =  -  ~i  +  j^  9-i  +  even  integral  powers  of  z. 

But,  in  the  vicinity  of  the  origin, 

,5-7  =  —  +  YQ^  -I-  even  integral  powers  of  2, 


132.]  OF   WEIEHSTRASS'S   FUNCTION  265 

so  that 

9P        /  d@      »/  \  dP      1  32lP  •  x  r 

17  3^-  +  V  g  :S  +  f<«)  |^t  i  gji  ™i**  even  mtegral  powers  of  z. 

The  function  on  the  left-hand  side  is  doubly-periodic  :  it  has  no  infinity 
at  the  origin  and  therefore  none  in  the  fundamental  parallelogram  ;  it  there 
fore  has  no  infinities  in  the  plane.  It  is  thus  constant  and  equal  to  its  value 
anywhere,  say  at  the  origin.  This  value  is  ^gz>  and  therefore 


T/w's  equation,  when  combined  with 


, 

+  eo'  ;      +  z        =  -  ty, 
dco          da>          oz 


j 

gives  the  value  of  ~-  and  -^  ,  . 
J  dm         9&) 

The  equations  are  identically  satisfied.     Equating  the  coefficients  of  z2  in 
the  expansions,  which  are  valid  in  the  vicinity  of  the  origin,  we  have 


and  equating  the  coefficients  of  ^  in  the  same  expansions,  we  have 


Hence  for  any  function  u,  which  involves  w  and  &/  and  therefore  implicitly 
involves  g2  and  ^r3,  we  have 

du        ,du 
w  5-  +  w  —  ,  = 

aw      a&) 

9w  .     ,  3w 
17  a-  +T;'  —  =  - 
9&)         9&) 

Since  ^)  is  such  a  function,  we  have 


f  :  *' 

being  ^/te  equations  which  determine  the  derivatives  of  $  with  regard  to  the 
invariants  g.,  and  g.^. 


266  EVEN  [132. 

The  latter  equation,  integrated  twice,  leads  to 
9V  da-      2       80-        1 


a  differential  equation  satisfied  by  <r(z)*. 

133.  The  foregoing  investigations  give  some  of  the  properties  of  doubly- 
periodic  functions  of  the  second  order,  whether  they  be  uneven  and  have  two 
simple  irreducible  infinities,  or  even  and  have  one  double  irreducible  infinity. 

If  a  function  U  of  the  second  order  have  a  repeated  infinity  at  z  =  y,  then 
it  is  determined  by  an  equation  of  the  form 


or,  taking  U  -  £  (X  +  fi  +  v)  =  Q,  the  equation  is 

Q'»  =  4a2  [(Q  -  e,)  (Q  -  e,)  (Q  -  $,)]*, 

where  ^  +  e2  +  e3  =  0.     Taking  account  of  the  infinities,  we  have 

Q=@(az-  ay)  ; 

and  therefore     U-±(\  +  /Ji.  +  v)  =  %>  (az  -  ay) 

.     .       1  (tp'(az)  +  cp  (ay)}'2 
=  -Q  (az)  -  <o  (ay)  +  —  — 

a    x      '        o    x     " 


by  Ex.  4,  p.  262.  The  right-hand  side  cannot  be  an  odd  function;  hence 
an  odd  function  of  the  second  order  cannot  have  a  repeated  infinity.  Similarly, 
by  taking  reciprocals  of  the  functions,  it  follows  that  an  odd  function  of  the 
second  order  cannot  have  a  repeated  zero. 

It  thus  appears  that  the  investigations  in  §§  120,  121  are  sufficient  for  the 
included  range  of  properties  of  odd  functions.  We  now  proceed  to  obtain 
the  general  equations  of  even  functions.  Every  such  function  can  (by  §  118, 
XIII.,  Cor.  I.)  be  expressed  in  the  form  |a#>  (z)  +b}+  {c#>  (z)  +  d],  and  its 
equations  could  thence  be  deduced  from  those  of  p(z)\  but,  partly  for 
uniformity,  we  shall  adopt  the  same  method  as  in  §  120  for  odd  functions. 
And,  as  already  stated  (p.  251),  the  separate  class  of  functions  of  the  second 
order  that  are  neither  even  nor  odd,  will  not  be  discussed. 

134.  Let,  then,  <j>(z)  denote  an  even  doubly-periodic  function  of  the 
second  order  (it  may  be  either  of  the  first  class  or  of  the  second  class)  and  let 
2&),  2<w'  be  its  periods ;  and  denote  2&)  +  2ft)'  by  2o>".  Then 

since  the  function  is  even ;  and  since 

<£  (ft)  +  Z)  =  <f>  (—  &)  —  z) 

=  <f>  (2&)  —  &)  —  z) 
=  (j)  (CD  —  *), 

*  For  this  and  other  deductions  from  these  equations,  see  Frobenius  und  Stickelberger,  Crelle, 
t.  xcii,  (1882),  pp.  311—327;  Halphen,  Traite  des  feme  t  ions  elliptiqucs,  t.  i,  (1886),  chap.  ix. ; 
and  a  memoir  by  the  author,  quoted  on  p.  254,  note. 


134.]  DOUBLY-PERIODIC   FUNCTIONS 

it  follows  that  <£  (&>  +  z)—  and,  similarly,  $  (&>'  +  z)  and  0  (to"  +  z)  are  even 
functions. 

Now  </>  (w  +  a),  an  even  function,  has  two  irreducible  infinities,  and  is 
periodic  in  2&>,  2&/  ;  also  <£  (z),  an  even  function,  has  two  irreducible  infinities 
and  is  periodic  in  2&>,  2&/.  There  is  therefore  a  relation  between  0  (z)  and 
</>  (w  +z),  which,  by  §  118,  Prop.  XIII.,  Cor.  I.,  is  of  the  first  degree  in  <£  (z)  and 
of  the  first  degree  in  <j)  (&>  +  z)  ;  thus  it  must  be  included  in 

B<f>  (z)  <j>(<o  +  z)-C<l>  (z)  -C'<t>(a>  +  z)+A  =  0. 

But  <£  (z)  is  periodic  in  2<w  ;  hence,  on  writing  z  +  <w  for  z  in  the  equation,  it 

becomes 

B<f>(a>+z)<j>(z)-C<f>(co+z)-C'<l>(z)  +  A=0; 

thus  tf=C". 

If  B  be  zero,  then  (7  may  not  be  zero,  for  the  relation  cannot  become 
evanescent  :  it  is  of  the  form 


A'  ..............................  (1). 

If  B  be  not  zero,  then  the  relation  is 


Treating  <f>  (w  +  z)  in  the  same  way,  we  find  that  the  relation  between  it 

and  (f)  (z)  is 

F(j>  (z)  (f>  (ay'  +  z)-D(j>  (z}  -D(j>(a>'  +  z)  +  E  =  0, 

so  that,  if  F  be  zero,  the  relation  is  of  the  form 

£(*)  +  0(a>'  +  *)  =  J0'  ...........................  (I)', 

and,  if  F  be  not  zero,  the  relation  is  of  the  form 


Four  cases  thus  arise,  viz.,  the  coexistence  of  (1)  with  (1)',  of  (1)  with  (2)', 
of  (2)  with  (1)',  and  of  (2)  with  (2)'.     These  will  be  taken  in  order. 

I.  :  the  coexistence  of  (1)  with  (1)'.     From  (1)  we  have 

<j>  (a>  +  z}  +  (j>  (&>"  +  z)  =  A', 

so  that  </>  (z)  +  <f)  (w  +  z)  +  (f)  (w  +  z)  +  0  (<w"  +  z)  =  2A'. 

Similarly,  from  (1)', 


so  that  A  =  E',  arid  then 

(f)((0  +  z)=(j)((o'+  Z\ 

whence  <w  ~  &>'  is  a  period,  contrary  to  the  initial  hypothesis  that  2&>  and  2&>' 
determine  a  fundamental  parallelogram.  Hence  equations  (1)  and  (1)'  cannot 
coexist. 


268  EVEN  [134. 

II.  :  the  coexistence  of  (1)  with  (2)'.     From  (1)  we  have 
<£(«"  +  z}  =  A'  -  <£(&>'  +  z) 


on  substitution  from  (2)'.     From  (2)'  we  have 


cb  (co   +  z)  =  -5*1)  --  (  —  =r 
F<f)  (CD  +  z)  -  D 

_  (A'D  -E)-D<j>  (z) 
=  A'F  -  D  -  F(f>  (z)  ' 
on  substitution  from  (1).     The  two  values  of  <£  (&>"  +  z)  must  be  the  same, 

whence 

A'F-D  =  D, 

which  relation  establishes  the  periodicity  of  </>  (z)  in  2ft)",  when  it  is  considered 
as  given  by  either  of  the  two  expressions  which  have  been  obtained.     We 

thus  have 

A'F=W- 
and  then,  by  (1),  we  have 

<f>(z)-j+<l>( 
and,  by  (2)',  we  have 


If  a  new  even  function  be  introduced,  doubly-periodic  in  the  same  periods 
having  the  same  infinities  and  defined  by  the  equation 

&  0)  =  </>  0)  -  J  > 
the  equations  satisfied  by  fa  (z)  are 

fa(a>  +  z)  +  fa(z)  =  0  } 

fa  (&)'  +  z)  fa  (z)  =  constant]  ' 

To  the  detailed  properties  of  such  functions  we  shall  return  later  ;  meanwhile 
it  may  be  noticed  that  these  equations  are,  in  form,  the  same  as  those  satisfied 
by  an  odd  function  of  the  second  order. 

III.  :  the  coexistence  of  (2)  with  (1)'.  This  case  is  similar  to  II.,  with  the 
result  that,  if  an  even  function  be  introduced,  doubly-periodic  in  the  same 
periods  having  the  same  infinities  and  defined  by  the  equation 

C 

fa  (Z)  =  <£  (2)  -  -g  , 

the  equations  satisfied  by  fa  (z)  are 

fa  (&>'  +  z)  +  fa  (z)  =  0  } 

fa  (&)  +  z)  fa  (z)  =  constant]  ' 

It  is,  in  fact,  merely  the  previous  case  with  the  periods  interchanged. 


134.]  DOUBLY-PERIODIC    FUNCTIONS  269 

IV.  :  the  coexistence  of  (2)  with  (2)'.     From  (2)  we  have 


_  (CD  -  AF)  <ft  (z)  -  (GE  -  AD) 
~  (BD  -  CF)  <J>  (z)  -  (BE  -  CD)  ' 

on  substitution  from  (2)'.     Similarly  from  (2)',  after  substitution  from  (2),  we 
have 


~ 


The  two  values  must  be  the  same  ;  hence 

CD-AF=-(GD-BE\ 

which  indeed  is  the  condition  that  each  of  the  expressions  for  <ft  (&>"  +  z) 
should  give  a  function  periodic  in  2&>".     Thus 


One  case  may  be  at  once  considered  and  removed,  viz.  if  C  and  D  vanish 
together.  Then  since,  by  the  hypothesis  of  the  existence  of  (2)  and  of  (2)', 
neither  B  nor  F  vanishes,  we  have 

A__E 

B~     F' 

so  that  u  +  ,  = 


and  then  the  relations  are     <£  (&>  +  z)  +  <f)  (&>'  +  z)  =  0, 

or,  what  is  the  same  thing,     <ft  (Y)  +  <ft  (&>"  +  z)  —  0  ] 

and  </>  (z)  </>(&>  +  z)  =  constant  j  ' 

This  case  is  substantially  the  same  as  that  of  II.  and  III.,  arising  merely 

from  a  modification  (§  109)  of  the  fundamental  parallelogram,  into  one  whose 

sides  are  determined  by  2&>  and  2&>". 

Hence  we  may  have  (2)  coexistent  with  (2)'  provided 

AF  +  BE=WD; 
C  and  D  do  not  both  vanish,  and  neither  B  nor  F  vanishes. 

IV.  (1).     Let  neither  C  nor  D  vanish  ;  and  for  brevity  write 

<f>((o  +  z)=<l>1,     (f>  (w"  +  z)  =  <£o,     </>  (&)'  +  z)  =  $3,     (f)  (z)  =  0. 
Then  the  equations  in  IV.  are 


Now  a  doubly-periodic  function,  with  given  zeros  and  given  infinities,  is 
determinate  save  as  to  an  arbitrary  constant  factor.  We  therefore  introduce 
an  arbitrary  factor  X,  so  that 

<£=Xi/r, 

G  D 

and  then  taking  =  CI'  =  Ca' 


270  EVEN  [134. 

£ 
we  have  (^  -  Cl)  (fa -  Cj)  =  d2-  -^ , 

ET 


The  arbitrary  quantity  A,  is  at  our  disposal  :  we  introduce  a  new  quantity  c2, 
defined  by  the  equation 

A 

Tt-.  o  —  Ci  (C2  +  €3)       C2C3  , 

and  therefore  at  our  disposal.     But  since 

AF  +  BE=2CD, 

A        E        .CD 

we  have  ^  +  ^  =  2  ^  ^  =  2Clc3j 

ri 

and  therefore  ^--2  =  c3  (Cj  +  c2)  -  0^2  . 

Hence  the  foregoing  equations  are 

-  d)  =  (Cj  -  C2)  (d  -  C3), 

-  C3)  =  (C3  -  d)  (C3  -  C2). 

The  equation  for  ^>2,  that  is  <f>((o"  +  z),  is 

_Lcf)-M 


where        L=  CD  -  BE  =  AF  -  CD,     M=AD-CE,     N=CF-BD, 

so  that  ^  +  5M"  =  2CL. 

As  before,  one  particular  case  may  be  considered  and  removed.     If  N  be 

zero,  so  that 

C_D_ 

B~F~a 

AE        CD     ,. 

say,  and  B+F=RF=       ' 

then  we  find  $  +  ^>2  =  ^>i  +  <£3  =  2«, 

or  taking  a  function  ^  =  0  —  a, 

the  equation  becomes         %  (^  +  %  C^"  +  ^  =  0. 
The  other  equations  then  become 


and  therefore  they  are  similar  to  those  in  Cases  II.  and  III. 
If  N  be  not  zero,  then  it  is  easy  to  shew  that 
N=BF\(c1-c3)> 


M  =  BF\3  (d  -  c3)  (c.,C!  +  c,c3  -  dc3)  ; 


134.]  DOUBLY-PERIODIC   FUNCTIONS  271 

and  then  the  equation  connecting  0  and  02  changes  to 


s  -  Ca)  =  (Ca  -  Cx)  (Ca  -  Cs) 

which,  with  (^  —  d)  ("^i  —  d)  =  (d  —  c2)  (d  —  c3) 

(  r  —  ^3/  \  i  3  —  ^3/  ==  V^3      ^"  ^3      ^2' 

are  relations  between  ty,  ^rl}  -^2,  ty.3,  where  the  quantity  c2  is  at  our  disposal. 

IV.  (2).  These  equations  have  been  obtained  on  the  supposition  that 
neither  G  nor  D  is  zero.  If  either  vanish,  let  it  be  C:  then  D  docs  not 
vanish ;  and  the  equations  can  be  expressed  in  the  form 

E 


D\ 

J 

E\         E(D*-EF) 


We  therefore  obtain  the  following  theorem  : 

If  (f>  be  an  even  function  doubly-periodic  in  2&>  and  2&>'  and  of  the  second 
order,  and  if  all  functions  equivalent  to  <J>  in  the  form  R<f>  +  8  (where  R  and 
S  are  constants)  be  regarded  as  the  same  as  0,  then  either  the  function  satisfies 
the  system  of  equations 


00)     0O" 
where  H  is  a  constant  ;  or  it  satisfies  the  system  of  equations 

{0  0)  -  d}  {0  (ft)  +Z)-  d]  =  (Ci  -  C2)  (d  -  C3) 
{00)-C3}{0(>/  +^)-C3}=(Cs-C1)(Cs-Ca) 
{0  0)  -  C2}  (0  ((,)"  +  Z)-  C2}  =  (C2  ~  d)  (C2  -  Cs) 

where  of  the  three  constants  clt  c2,  cs  one  can  be  arbitrarily  assigned. 
We  shall  now  very  briefly  consider  these  in  turn. 

135.     So  far  as  concerns  the  former  class  of  equations  satisfied  by  an  even 
doubly-periodic  function,  viz., 


we  proceed  initially  as  in  (§  120)  the  case  of  an  odd  function.     We  have  the 
further  equations 

00)  =  0(-4 

0  (ft)  +  Z)  —  0  (ft)  —  Z),       0  (a/  +  Z)  =  0  (ft)'  —  Z). 

*  The  systems  obtained  by  the  interchange  of  w,  w',  w"  among  one  another  in  the  equations 
are  not  substantially  distinct  from  the  form  adopted  for  the  system  I.  ;  the  apparent  difference 
can  be  removed  by  an  appropriate  corresponding  interchange  of  the  periods. 


272  EVEN   DOUBLY-PERIODIC   FUNCTIONS  [135. 

Taking  z  =  —  ^w,  the  first  gives 


so  that  ^&>  is  either  a  zero  or  an  infinity. 
If  \<£>  be  a  zero,  then 

(f>  (f  to)  =  $  (<«  +  ^ft))  =  —  <f>  (^»)  by  the  first  equation 

=  0, 
so  that  ^&>  and  f&>  are  zeros.     And  then,  by  the  second  equation, 

&)'  +  ^<w,     &)'  4-  f  a) 
are  infinities. 

If  \w  be  an  infinity,  then  in  the  same  way  |w  is  also  an  infinity  ;  and 
then  a)'  +  \w,  &>'  +  f  &)  are  zeros.  Since  these  amount  merely  to  interchanging 
zeros  and  infinities,  which  is  the  same  functionally  as  taking  the  reciprocal  of 
the  function,  we  may  choose  either  arrangement.  We  shall  take  that  which 
gives  ^0),  f  &>  as  the  zeros  ;  and  &>'  4-  ^&>,  &/  +  f  &>  as  the  infinities. 

The  function  <j>  is  evidently  of  the  second  class,  in  that  it  has  two  distinct 
simple  irreducible  infinities. 

Because  &>'  +  |&),  &>'  +  f  &>  are  the  irreducible  infinities  of  </>  (z),  the  four 
zeros  of  $'  (z}  are,  by  §  117,  the  irreducible  points  homologous  with  &>", 
&)"  +  &>,  &>"  +  a)',  a)"  +  &)",  that  is,  the  irreducible  zeros  of  (f)'  (z)  are  0,  &>,  &>',  &>". 
Moreover 


by  the  first  of  the  equations  of  the  system  ;  hence  the  relation  between  (f>  ( 
and  ((>'  (z)  is 

#*  (z}  =  A{<t>(z)-$  (())}  {</>  (z)  -  (/>  («)}  |0  (*)  -  (/>  (ft)')}  {(/>  (*)  -  </>  («")} 

=  A  [p  (0)  -  p  (z)}{p  (ft)')  -  ^  (*)}. 
Since  the  origin  is  neither  a  zero  nor  an  infinity  of  <£  (^),  let 


so  that  </>j  (0)  is  unity  and  0/  (0)  is  zero  ;  then 

^(*)«X»{l-^(*)}{^-^(f)) 

the  differential  equation  determining  fa  (z). 

The  character  of  the  function  depends  upon  the  value  of  p  and  the 
constant  of  integration.     The  function  may  be  compared  with  en  u,  by  taking 

2ft),   2&/  =  4>K,  2K  +  2iK'  ;    and  with    —  *—  ,  by  taking  2ft),  2ft)'  =  2K,  MK', 

dn  u 

which  (§  131,  note)  are  the  periods  of  these  (even)  Jacobian  elliptic  functions. 
We  may  deal  even  more  briefly  with  the  even,  function  characterised  by 
the  second  class  of  equations  in  §  134.     One  of  the  quantities  c1}  c2,  c3  being 
at  our  disposal,  we  choose  it  so  that 

Ci  +  c2  +  c3  =  0  ; 

and  then  the  analogy  with   the  equations   of  Weierstrass's    ^-function   is 
complete  (see  §  133). 


CHAPTER   XII. 

PSEUDO-PERIODIC  FUNCTIONS. 

136.  MOST  of  the  functions  in  the  last  two  Chapters  are  of  the  type 
called  doubly-periodic,  that  is,  they  are  reproduced  when  their  arguments  are 
increased  by  integral  multiples  of  two  distinct  periods.  But,  in  §§  127,  130, 
functions  of  only  a  pseudo-periodic  type  have  arisen :  thus  the  ^-function 
satisfies  the  equation 

m2&>  +  m'2&>')  =  £(»  +  m2i)  +  m'2v', 
,nd  the  cr-function  the  equation 

m'     i  (mr,+m'r,')  (z+wuo+m'oi1) 


These  are  instances  of  the  most  important  classes:  and  the  distinction 
between  the  two  can  be  made  even  less  by  considering  the  function 
e^(z}  —  ^(z),  when  we  have 

£  (z  +  ra2&>  +  m'2&>')  =  e-mr>  e"™'*'  %  (z). 

In  the  case  of  the  ^-function  an  increase  of  the  argument  by  a  period  leads 
to  the  reproduction  of  the  function  multiplied  by  an  exponential  factor  that 
is  constant,  and  in  the  case  of  the  <r-function  a  similar  change  of  the 
argument  leads  to  the  reproduction  of  the  function  multiplied  by  an 
exponential  factor  having  its  index  of  the  form  az  +  b. 

Hence,  when  an  argument  is  subject  to  periodic  increase,  there  are  three 
simple  classes  of  functions  of  that  argument. 

First,  if  a  function  f(z)  satisfy  the  equations 

/(*  +  2fi>)  =/(*),    /(*  +  2«')  =/(*), 

it  is  strictly  periodic :  it  is  sometimes  called  a  doubly-periodic  function  of  the 
first  kind.  The  general  properties  of  such  functions  have  already  been 
considered. 

Secondly,  if  a  function  F(z)  satisfy  the  equations 

F  (z  +  2&>)  =  pF  (z),     F  (z  +  2&/)  -  pfF  (z), 
F-  18 


274  THREE    KINDS  [136. 

where  /u,  and  fjf  are  constants,  it  is  pseudo-periodic  :  it  is  called  a  doubly- 
periodic  function  of  the  second  kind.  The  first  derivative  of  the  logarithm 
of  such  a  function  is  a  doubly-periodic  function  of  the  first  kind. 

Thirdly,  if  a  function  <f>  (z)  satisfy  the  equations 

<j>(z  +  2o))  =  eaz+b  <j>  (z\     <f>(z  +  2ft)')  =  ea'z+v  (j>  (z), 

where  a,  b,  a',  b'  are  constants,  it  is  pseudo-periodic  :  it  is  called  a  doubly- 
periodic  function  of  the  third  kind.  The  second  derivative  of  the  logarithm 
of  such  a  function  is  a  doubly-periodic  function  of  the  first  kind. 

The   equations   of  definition   for   functions   of   the   third   kind   can   be 
modified.     We  have 

.  <f>  (Z  +  2ft)  +  2ft)')  =  e«(2+2<o')+6+a'z+6'  <£  (z) 
—  ga'  (2+2o>)  +b'+az+b  J,  fz\ 

whence  a'oo  —  am'  =  —  nnri, 

where  ra  is  an  integer.     Let  a  new  function  E  (z)  be  introduced,  defined  by 

the  equation 

£(«)«*"+*•  t(*)i 

then  X  and  /A  can  be  chosen  so  that  E  (z}  satisfies  the  equations 

E(z  +  2a))  =  E  (z\     E(z+  2ft)')  =  eAz+£  E  (z\ 
From  the  last  equations,  we  have 

E  (z  +  2&)  +  2ft)')  =  eA(*+**+B  E  ^ 

=  eAz+s  E  (z), 
so  that  2Aa)  is  an  integral  multiple  of  2?™'. 

Also  we  have       E(z  +  2o>)  =  e*(*-*»'+^+a»)  <j>(z  +  2o>) 


so  that  4X&)  +  a  =  0, 

and  4A,ftr  +  2/A&)  +6  =  0  (mod. 

Similarly,  E  (z  +  2ft)')  =  e^+wj'+^+w,  0  ^  +  2ft)') 


so  that  4Xo)'  +  a  =  A, 

and  4W2  +  2/^co'  +  6'  =  B  (mod.  27ri). 

From  the  two  equations,  which  involve  X  and  not  //,,  we  have 

Aco  =  a'o)  —  aw' 


agreeing  with  the  result  with  2  A  co  is  an  integral  multiple  of  Ziri. 

And  from  the  two  equations,  which  involve  /j,,  we  have,  on  the  elimination 
of  /j,  and  on  substitution  for  X  and  A, 

b'co  —  6ft)'  —  a&)'  (ft)'  —  &))  =  5ft)  (mod.  2-Tn'). 


136.]  OF   DOUBLY-PERIODIC   FUNCTIONS  275 

If  A  be  zero,  then  E(z)  is  a  doubly-periodic  function  of  the  first  kind 
when  eB  is  unity,  and  it  is  a  doubly-periodic  function  of  the  second  kind 
when  eB  is  not  unity.  Hence  A,  and  therefore  m,  may  be  assumed  to  be 
different  from  zero  for  functions  of  the  third  kind.  Take  a  new  function 
3?z  such  that 


mm 
then  <l>  (z)  satisfies  the  equations 


4)  (z  +  2&))  =  <I>  (z\     <&(z  +  2o)')  =  e     w     3>(z) 

*  /  \     /'  \  /  \     /t 

which  will  be  taken  as  the  canonical  equations  defining  a  doubly -periodic 
function  of  the  third  kind. 

Ex.     Obtain  the  values  of  X,  p,  A,  B  for  the  Weierstrassian  function  ir(z). 

We  proceed  to  obtain  some  properties  of  these  two  classes  of  functions 
which,  for  brevity,  will  be  called  secondary-periodic  functions  and  tertiary- 
periodic  functions  respectively. 

Doubly-Periodic  Functions  of  the  Second  Kind. 

For  the  secondary-periodic  functions  the  chief  sources  of  information  are 

Hermite,  Comptes  Rendus,  t.  liii,  (1861),  pp.  214—228,  ib.,  t.  Iv,  (1862),  pp.  11—18, 
85 — 91  ;  Sur  quelques  applications  des  fonctions  elliptiques,  §§  I — in,  separate 
reprint  (1885)  from  Comptes  Rendus;  "Note  sur  la  theorie  des  fonctions  ellip 
tiques"  in  Lacroix,  vol.  ii,  (6th  edition,  1885),  pp.  484—491;  Cours  d' Analyse, 
(4me  ed.),  pp.  227—234. 

Mittag-Leffler,  Comptes  Rendus,  t.  xc,  (1880),  pp.  177 — 180. 

.Frobenius,  Crelle,  t.  xciii,  (1882),  pp.  53 — 68. 

Brioschi,  Comptes   Rendus,  t.  xcii,  (1881),  pp.  325—328. 

Halphen,  Traite'  des  fonctions  elliptiques,  t.  i,  pp.  225 — 238,  411 — 426,  438 442,  463. 

137.  In  the  case  of  the  periodic  functions  of  the  first  kind  it  was  proved 
that  they  can  be  expressed  by  means  of  functions  of  the  second  order  in  the 
same  period — these  being  the  simplest  of  such  functions.  It  will  now  be 
proved  that  a  similar  result  holds  for  secondary- periodic  functions,  defined  by 
the  equations 


Take  a  function  Q  (z}  = 


a  (z)  a-  (a) 

then  we  have  G(z+2a>)  =  <r(*  +  g 

a  (a)  a  (z  +  2w) 


arid  G(z+  2&/)  =  e'V«+2W  Q.  (^). 

The  quantities  a  and  X  being  unrestricted,  we  choose  them  so  that 

„  _  g2rja+2A<o  '  __  g2T)'a+2A(o'  • 

and  then  G  (z),  a  known  function,  satisfies  the  same  equation  as  F  (z). 

18—2 


276  PSEUDO-PERIODIC   FUNCTIONS  [137. 

Let  u  denote  a  quantity  independent  of  z,  and  consider  the  function 

f(Z)  =  F(z)G(u-z}. 
We  have  f(z  +  2o>)  =  F(z  +  2o>)  G  (u-  z  -  2w) 


=/(*)  ; 

and  similarly  f(z  +  2<o')  =f(z), 

so  that/(X)  is  a  doubly-periodic  function  of  the  first  kind  with  2«  and  2o>' 

for  its  periods. 

The  sum  of  the  residues  of  f(z)  is  therefore  zero.  To  express  this  sum, 
we  must  obtain  the  fractional  part  of  the  function  for  expansion  in  the 
vicinity  of  each  of  the  (accidental)  singularities  of  f(z),  that  lie  within  the 
parallelogram  of  periods.  The  singularities  of/  (2)  are  those  of  G  (u  —  z)  and 
those  of  F(z). 

Choosing  the  parallelogram  of  reference  so  that  it  may  contain  u,  we  have 
z  =  u  as  the  only  singularity  of  G  (u  —  z)  and  it  is  of  the  first  order,  so  that, 
since 

$(£)  —  =+  positive  integral  powers  of  f 
in  the  vicinity  of  £=  0,  we  have,  in  the  vicinity  of  u, 
f(z)  =  {F  (u)  +  positive  integral  powers  of  u  —  z}  \  —    -4-  positive  powers  I 

=  --  —  +  positive  integral  powers  of  z  —  u  ; 

hence  the  residue  of/(Y)  for  u  is  —F(u}. 

Let  z  =  c  be  a  pole  of  F  (z)  in  the  parallelogram  of  order  n  +  1  ;  and,  in 
the  vicinity  of  c,  let 

(?!        _  cf  /    1   \  „       dn  (    1    \ 

F(z)  =  ^—c  +G^Z  (jr^J  +  •  •  •  +  C'n+i  fan  (zITc)  +  P°sltlve  integral  powers. 

Then  in  that  vicinity 


and  therefore  the  coefficient  of  -        in  the  expansion  of  f(z)  for  points  in  the 

Z  ~~  0 

vicinity  of  c  is 


which  is  therefore  the  residue  off(z)  for  c. 

This  being  the  form  of  the  residue  of  f(z)  for  each  of  the  poles  of  F  (z), 
then,  since  the  sum  of  the  residues  is  zero,  we  have 


137.]  OF   THE   SECOND    KIND  277 

or,  changing  the  variable, 


,.  ..n+l     n         - 

where  the  summation  extends  over  all  the  poles  of  F(z)  within  that  parallelo 
gram  of  periods  in  which  z  lies.  This  result  is  due  to  Hermite. 

138.  It  has  been  assumed  that  a  and  \,  parameters  in  0,  are  determinate, 
an  assumption  that  requires  /j,  and  ^  to  be  general  constants  :  their  values 
are  given  by 

yd  4-  &>X  =  |  log  fjb,     r)'a  +  &/X  —  \  log  //, 

and,  therefore,  since  ijca'  —  rfca  =  ±  ^ITT,  we  have 

+  ITTCL  =     w'  log  /JL  —  co  log  //) 

+  iir\  =  —  V)  log  /i  +  77  log  /z'j  ' 

Now  X  may  vanish  without  rendering  G  (z)  a  null  function.  If  a  vanish  (or, 
what  is  the  same  thing,  be  an  integral  combination  of  the  periods),  then  G  (z) 
is  an  exponential  function  multiplied  by  an  infinite  constant  when  X  does  not 
vanish,  and  it  ceases  to  be  a  function  when  X  does  vanish.  These  cases  must 
be  taken  separately. 

First,  let  a  and  X  vanish*  ;  then  both  //,  and  ///  are  unity,  the  function  F 
is  doubly-periodic  of  the  first  kind  ;  but  the  expression  for  j^is  not  determinate, 
owing  to  the  form  of  G.  To  render  it  determinate,  consider  X  as  zero  and  a 
as  infinitesimal,  to  be  made  zero  ultimately.  Then 

„,,      o-(z)  +  aa'(z)  +  ...  .^ 

(*(z)  =  -      -  ~  —     —  (1  +  positive  integral  powers  of  a) 

=  -  +  £  (z)  +  positive  powers  of  a. 
a 

Since  a  is  infinitesimal,  /JL  and  /j,'  are  very  nearly  unity.  When  the 
function  F  is  given,  the  coefficients  C1}  <72,  ...  may  be  affected  by  a,  so  that 
for  any  one  we  have 

Ck  —  bk  +  ayk  +  higher  powers  of  a, 

where  yh  is  finite  ;  and  bk  is  the  actual  value  for  the  function  which  is  strictly 
of  the  first  kind,  so  that 

Sk-O, 

the  summation  being  extended  over  the  poles  of  the  function.  Then  retaining 
only  a"1  and  a°,  we  have 


This  case  is  discussed  by  Hermite  (I.e.,  p.  275). 


278  MITTAG-LEFFLER'S  THEOREM  [138. 

where  C0,  equal  to  £71,  is  a  constant  and  the  term  in  -  vanishes.    This  expres- 

CL 

sion,  with  the  condition  S^  =  0,  is  the  value  of  F  (u)  or,  changing  the  variables, 
we  have 


with  the  condition  S&i  =  0,  a  result  agreeing  with  the  one  formerly  (§  128) 
obtained. 

When  F  is  not  given,  but  only  its  infinities  are  assigned  arbitrarily,  then 
SO  =  0  because  F  is  to  be  a  doubly-periodic  function  of  the  first  kind  ;  the 

term  -  "£C  vanishes,  and  we  have  the  same  expression  for  F(z)  as  before. 
Secondly,  let  a  vanish*  but  not  \,  so  that  ^  and  //  have  the  forms 


We  take  a  function  g  (z)  = 

then  g(z-  2o>)  =  ^  e^  £  (z  -  2eo  ) 


and  g(z-2a>')  =  p'-1  {g  (z}  -  2?/  e^}  . 

Introducing  a  new  function  H  (z)  defined  by  the  equation 


we  have  H  (z  +  2t»)  =  H  (z)  -  2ijeA  <«-*»  F  (z), 

and  H  (z  +  2o>')  =  H  (z)  -  27?V<M-*>  F(z). 

Consider  a  parallelogram  of  periods  which  contains  the  point  u  ;  then,  if  ©  be 

the  sum  of  the  residues  of  H  (z)  for  poles  in  this  parallelogram,  we  have 


the  integral  being  taken  positively  round  the  parallelogram.     But,  by  §  116, 
Prop.  II.  Cor.,  this  integral  is 


f  e-*(p+*-«)  F  (p  +  2tot)  dt  -  0/77  f  e-^+ 
Jo  Jo 


where  p  is  the  corner  of  the  parallelogram  and  each  integral  is  taken  for  real 
values  of  t  from  0  to  1.  Each  of  the  integrals  is  a  constant,  so  far  as  concerns 
u  ;  and  therefore  we  may  take 

®  =  -Ae^u, 

the  quantity  inside  the  above  bracket  being  denoted  by  —\i-rrA. 

The  residue  of  H  (z)  for  z  =  u,  arising  from  the  simple  pole  of  g  (u  —  z),  is 
-F(u)  as  in§  137. 

If  z  =  c  be  an  accidental  singularity  of  F  (z)  of  order  n+1,  so  that,  in  the 
vicinity  of  z  =  c, 

F(.)  =  C,          +  0.        A-    +  .  .  .  +  BU  i-   +  P  (,  -  c), 


This  is  discussed  by  Mittag-Leffler,  (I.e.,  p.  275). 


138.]  ON    SECONDARY    FUNCTIONS  279 

then  the  residue  of  H  (z)  for  z  =  c  is 

d  dn 


and  similarly  for  all  the  other  accidental  singularities  of  F  (z}.     Hence 


F(z)  =  A**  + 

where  the  summation  extends  over  all  the  accidental  singularities  of  F  (z)  in  a 
parallelogram  of  periods  which  contains  z,  and  y  (z)  is  the  function  exz%(z}. 
This  result  is  due  to  Mittag-Leffler. 

Since  /*  =  e2*"  and 

g  (z  -  c  +  2&>)  =  fig  (z  -  c)  + 
we  have 


and  therefore  2  (Gl  +  C.2\  4-  . . .  +  Gn+l\n)  e~^  =  0, 

the  summation  extending  over  all  the  accidental  singularities  of  F(z).     The 

same  equation  can  be  derived  through  ^F(z)  =  F(z  +  2&>'). 

Again  2(7:  is  the  sum  of  the  residues  in  a  parallelogram  of  periods,  and 
therefore 


the  integral  being  taken  positively  round  it.     If  p  be  one  corner,  the  integral 

n 

F  (p  +  2co't)  dt, 

Jo 


IS 

/•i  n 


o 
,    each  integral  being  for  real  variables  of  t. 

Hermite's  special  form  can  be  derived  from  Mittag-Leffler's  by  making  \ 
vanish. 

Note.  Both  Hermite  and  Mittag-Leffler,  in  their  investigations,  have 
used  the  notation  of  the  Jacobian  theory  of  elliptic  functions,  instead  of 
dealing  with  general  periodic  functions.  The  forms  of  their  results  are  as 
follows,  using  as  far  as  possible  the  notation  of  the  preceding  articles. 

I.     When  the  function  is  denned  by  the  equations 

F  (z  +  2K)  =  ^F  (z),     F(z+  2iK')  =  ^F  (z), 

then  F(z)  = 


280  INFINITIES  AND   ZEROS  [138. 

(the  symbol  H  denoting  the  Jacobian  .ff-function),  and  the  constants  <w  and  X 
are  determined  by  the  equations 


II.     If  both  X  and  to  be  zero,  so  that  F(z)  is  a  doubly-periodic  function 
of  the  first  kind,  then 


with  the  condition  5$i  =  0. 

III.     If  W  be  zero,  but  not  X,  then 


... 
where  g  (z}  =  --&  V, 


the  constants  being  subject  to  the  condition 

2  (G,  +  C,\  +  .  .  .  +  Gn+1  X")e-Ac  =  0, 

and  the  summations  extending  to  all  the  accidental  singularities  of  F(z)  in  a 
parallelogram  of  periods  containing  the  variable  z. 

139.     Reverting  now  to  the  function  F(z)  we  have  G  (z),  defined  as 


a  (z)  a  (a) 

when  a  and  X  are  properly  determined,  satisfying  the  equations 
G(z  +  2a>)  =  ftG  (z),     £0  +  2&/)  =  yu/£0). 

Hence  H  (z)  =  F(z)/G  (z)  is  a  doubly-periodic  function  of  the  first  kind  ;  and 
therefore  the  number  of  its  irreducible  zeros  is  equal  to  the  number  of  its 
irreducible  infinities,  and  their  sums  (proper  account  being  taken  of  multipli 
city)  are  congruent  to  one  another  with  moduli  2«  and  2&>'. 

Let  Ci,  c2,...,  cm  be  the  set  of  infinities  of  F  (z)  in  the  parallelogram  of 
periods  containing  the  point  z  ;  and  let  y:,  .  .  .  ,  7^  be  the  set  of  zeros  of  F  (z)  in 
the  same  parallelogram,  an  infinity  of  order  n  or  a  zero  of  order  n  occurring 
n  times  in  the  respective  sets.  The  only  zero  of  0  (z)  in  the  parallelogram  is 
congruent  with  —  a,  and  its  only  infinity  is  congruent  with  0,  each  being 
simple.  Hence  the  m+l  irreducible  infinities  of  H  (z)  are  congruent  with 

a,  GI,  GZ,  .  .  .  ,  cm, 
and  its  /*  +  1  irreducible  zeros  are  congruent  with 

0,  71,  7s>  •••>%*; 
and  therefore  m  +  1  =  p,  +  1, 


139.]  OF   SECONDARY    FUNCTIONS  281 

From  the  first  it  follows*  that  the  number  of  infinities  of  a  doubly-periodic 
function  of  the  second  kind  in  a  parallelogram  of  periods  is  equal  to  the  number 
of  its  zeros,  and  that  the  excess  of  the  sum  of  the  former  over  the  sum  of  the 
latter  is  congruent  with 

,  (°>'  i              w  i        , 
+    — ,  log  it -.  log  u, 

-    \7Tl  TTl       6  ' 

/ 

the  sign  being  the  same  as  that  of  9t  ( — 

\10) 

The  result  just  obtained  renders  it  possible  to  derive  another  expression 
for  F  (z),  substantially  due  to  Hermite.  Consider  a  function 

F  (Z)  =  °-Q-7i)  0-0-72).. -0-0-7™) ePZ 

(T(z-c1)(r(z-c2)...ar(z-cm)       ' 

where  p  is  a  constant.  Evidently  F1  (z)  has  the  same  zeros  and  the  same 
infinities,  each  in  the  same  degree,  as  F  (z).  Moreover 

F,  (Z  +  2ft))  =  Fl  (Z)  e2,(2C-2y)  +  2pWj 
F1  (Z  +  2ft)')  =  F!  (Z)  e2V(2e-2y)+2P«,'t 

If,  then,  we  choose  points  c  and  7,  such  that 

Sc  —  £7  =  a, 
and  we  take  p  =  \  where  a  and  X  are  the  constants  of  G  (z),  then 

F,  (z  +  2co)  =  ^  (z),     F,  (z  +  2ft>')  =  n'Fj.  (z). 

The  function  Fl  (z)/F(z)  is  a  doubly-periodic  function  of  the  first  kind  and  by 
the  construction  of  Fl  (z)  it  has  no  zeros  and  no  infinities  in  the  finite  part  of 
the  plane:  it  is  therefore  a  constant.  Hence 

F(z]  =  A  gfr-'ftM*- •/»)•••*  (*—*») ^ 

a(z-  c,)  a-  (z  -  C2). .  .o-  (z  -  Cm) 

where  Sc  —  £7  =  a,  and  a  and  A,  are  determined  as  for  the  function  G  (z}. 

140.  One  of  the  most  important  applications  of  secondary  doubly-periodic 
functions  is  that  which  leads  to  the  solution  of  Lame's  equation  in  the  cases 
when  it  can  be  integrated  by  means  of  uniform  functions.  This  equation  is 
subsidiary  to  the  solution  of  the  general  equation,  characteristic  of  the 
potential  of  an  attracting  mass  at  a  point  in  free  space;  and  it  can  be 
expressed  either  in  the  form 

jY  =  (Ak'2  sn2  z  +  B)  w, 
or  in  the  form  -    2 -  =  (A@  (z)  +  B}  w, 

*  Frobenius,  Crelle,  xciii,  pp.  55 — 68,  a  memoir  which  contains  developments  of  the  properties 
of  the  function  G  (z).  The  result  appears  to  have  been  noticed  first  by  Brioschi,  (Comptes  Ilendus, 
t.  xcii,  p.  325),  in  discussing  a  more  limited  form. 


282  LAMP'S  [140. 

according  to  the  class  of  elliptic  functions  used.  In  order  that  the  integral 
may  be  uniform,  the  constant  A  must  be  n  (n  -f  1),  where  n  is  a  positive 
integer  ;  this  value  of  A,  moreover,  is  the  value  that  occurs  most  naturally  in 
the  derivation  of  the  equation.  The  constant  B  can  be  taken  arbitrarily. 

The  foregoing  equation  is  one  of  a  class,  the  properties  of  which  have 
been  established*  by  Picard,  Floquet,  and  others.  Without  entering  into 
their  discussion,  the  following  will  suffice  to  connect  them  with  the  secondary 
periodic  function. 

Let  two  independent  special  solutions  be  g  (z)  and  h  (z),  uniform  functions 
of  z  ;  every  solution  is  of  the  form  ag  (z}  +  /3h  (z},  where  a  and  /3  are  constants. 
The  equation  is  unaltered  when  z  +  2w  is  substituted  for  z  ;  hence  g  {z  +  2&>) 
and  h  (z  +  2&>)  are  solutions,  so  that  we  must  have 

g  (z  +  2w)  =  Ag  (z}  +  Bh  (z},     h(z  +  2o>)  =  Cg  (z)  +  Dh  (z\ 

where,  as  the  functions  are  determinate,  A,  B,  C,  D  are  determinate  constants, 
such  that  AD  —  BC  is  different  from  zero. 

Similarly,  we  obtain  equations  of  the  form 

g  (z  +  2co')  =  A'g  (z)  +  B'h  (z\     h(z  +  2co')  =  C'g  (z}  +  D'h  (z}. 
Using  both  equations  to  obtain  g  (z  +  2o>  +  2&/)  in  the  same  form,  we  have 

BC'  =  B'C,    AB'  +  BD'  =  A'B  +  B'D  ; 
and  similarly,  for  h  (z  +  2w  +  20)'),  we  have 


C     G'          A-D     A'-U 

therefore  -~  =  -™  =  o,  —  ~  —  =  —  ™  —  =  e. 

x>      -D  n  n 

Let  a  solution  F  (z}  =  ag  (z)  +  bh  (z) 

be  chosen,  so  as  to  give 


if  possible.     The  conditions  for  the  first  are 


a  b 

so  that  a/b  (=  £)  must  satisfy  the  equation 

and  the  conditions  for  the  second  are 

aA'  +  bCf     aB'  +  bD' 


*  Picard,  Comptes  Rendus,  t.  xc,  (1880),  pp.  128—131,  293—295;  Crelle,  t.  xc,  (1880),  pp. 
281—302. 

Floquet,  Comptes  Rendus,  t.  xcviii,  (1884),  pp.  82  —  85  ;  Ann.  de  VEc.  Norm.  Sup.,  3mc  Ser., 
t.  i,  (1884),  pp.  181—238. 


140.]  DIFFERENTIAL   EQUATION  283 

so  that  £  must  satisfy  the  equation 

A'-D'=^B'~~. 
These  two  equations  are  the  same,  being 

p.-«g-ft*a 

Let  £j  and  £2  be  the  roots  of  this  equation  which,  in  general,  are  unequal  ;  and 
let  fa,  fa  and  fa,  fa.'  be  the  corresponding  values  of  /z,  //.  Then  two  functions, 
say  FI  (z)  and  F^  (z),  are  determined  :  they  are  independent  of  one  another,  so 
therefore  are  g  (z)  and  h  (z)  ;  and  therefore  every  solution  can  be  expressed  in 
terms  of  them.  Hence  a  linear  differential  equation  of  the  second  order,  having 
coefficients  that  are  doubly-periodic  functions  of  the  first  kind,  can  generally  be 
integrated  by  means  of  doubly  -periodic  functions  of  the  second  kind. 

It  therefore  follows  that  Lame's  equation,  which  will  be  taken  in  the  form 


can  be  integrated  by  means  of  secondary  doubly-periodic  functions. 

141.     Let  z  =  c  be  an  accidental  singularity  of  w  of  order  m ;  then,  for 
points  z  in  the  immediate  vicinity  of  c,  we  have 


and  therefore 


2mp 

~  z-  c  +  P°SltlVe  P°wers  °f  *  - 


Since  this  is  equal  to  n  (n  +  1)  @  (z)  +  B 

it  follows  that  c  must  be  congruent  to  zero  and  that  m,  a  positive  integer, 
must  be  n.  Moreover,  p  =  0.  Hence  the  accidental  singularities  of  w  are 
congruent  to  zero,  and  each  is  of  order  n. 

The  secondary  periodic  function,  which  has  no  accidental  singularities 
except  those  of  order  n  congruent  to  z  =  0,  has  n  irreducible  zeros.  Let  them 
be  —  alt  —  a2,...,  —  an;  then  the  form  of  the  function  is 


Hence  1  *?  =  ,-»?«  + 


or,  taking  p  =  -  ^(ar),  we  have 


and  therefore         i  *?  -  1  (*?)' .  n(>  (,)  -  X  f>  («  +  «, 
19  O^      W2\dzj         *  v  y      r»i 


284  INTEGRATION 

But,  by  Ex.  3,  §  131,  we  have 


[141. 


4   r=1  £>  (ar)  -  p  (z) 


, 


by  Ex.  4,  §  131.     Thus 


W 

Now 


r=l  S=l 


. 

g>  (a.)  - 


g>  (ar)  -  g)  («)  '  g>  (a,)  -  g>  (^) 

4^?3  (^r)  -  ^2ip  Q)  -  #,  +  %>'  (a*)  &  (a,) 


where 


g>  (ar)  -  £>  (a.) 
Let  the  constants  a  be  such  that 


(O  -  £>  (a2) 


+ 


-H...-0 


/i  equations  of  which  only  n  —  1  are  independent,  because  the  sum  of  the  n 
left-hand  sides  vanishes.     Then  iu  the  double  summation  the  coefficient  of 

i      f  .1      f     u         #>'  (ar)  —  &'  (z)  . 
each  of  the  tractions  *   )—,-  —  V\  is  zero  ;  and  so 


and  therefore  •  -^-,  =  w  (w  +  1)  p  (z)  +  (2n  —  1)  2  ^>  (a,.). 

/IU   GLZ"  T=l 

Hence  it  follows  that 

_<T(z  +  aJ <T(z  +  a2)...<r(z  +  an)    -z?J("r) 
an  (z} 

satisfies  Lame's  equation,  provided  the  n  constants  a  be  determined  by  the 
preceding  equations  and  by  the  relation 

B  =  (2n-l)  I  pfa). 


141.]  OF  LAMP'S  EQUATION  285 

Evidently  the  equation  is  unaltered  when  —  z  is  substituted  for  z  ;  and 
therefore 


is  another  solution.     Every  solution  is  of  the  form 

MF(z}  +  NF(-z), 
where  M  and  N  are  arbitrary  constants. 

COEOLLARY.     The  simplest  cases  are  when  n  =  l  and  n  =  2. 

When  n  =  1,  the  equation  is 


•  j-r-  +  B  : 

w  dzz 

there  is  only  a  single  constant  a  determined  by  the  single  equation 

B  =  p  (a), 
and  the  general  solution  is 

,,  a  (2  +  a)        ...       ,ra(z  —  a]     ...  , 
w  =  M  —  ^-  7-^-/  e~2£(a)  +  N  -  ----  '  '  es^a> 
o-  (z}  a  (z) 

When  n  =  2,  the  equation  is 


-J-.  =  6(0  (z}  +  B. 

w  dz* 

The  general  solution  is 


^ 


where  a  and  b  are  determined  by  the  conditions 


Rejecting  the  solution  a+b  =  0,  we  have  a  and  b  determined  by  the  equations 
p  (a) 


For  a  full  discussion  of  Lame's  equation  and  for  references  to  the  original  sources  of 
information,  see  Halphen,  Traite  des  fonctions  elliptiques,  t.  ii,  chap,  xn.,  in  particular, 
pp.  495  et  seq. 

Ex.     When  Lamp's  equation  has  the  form 

1  d?w 
-  -T-5  =n  (n  +  1)  £2  sn20  -  h. 

w  dz2        ^ 

obtain  the  solution  for  w  =  l,  in  terms  of  the  Jacobian  Theta-Functions, 


where  co  is  determined  by  the  equation  dn2o>  =  A-F  ;  and  discuss  in  particular  the  solution 
when  h  has  the  values  l+£2,  1,  £2. 

Obtain  the  solution  for  »  =  2  in  the  form 


i  +B  -  fe^)e-  K&  .1 

J      SI   e(»)  j' 


286  PSEUDO-PERIODIC   FUNCTIONS  [141. 

where  X  and  w  are  given  by  the  equations 

(2P  sn2  a  -  1  -  F)  (2F  sn2  a  -  1)  (2  sn2  a  -  1) 
3Fsn4a-2(l+£2)sn2a  +  l  ~  ' 


and  a  is  derived  from  h  by  the  relation 


Deduce  the  three  solutions  that  occur  when  X  is  zero,  and  the  two  solutions  that  occur 
when  X  is  infinite.  (Hermite.) 

Doubly-Periodic  Functions  of  the  Third  Kind. 

142.     The  equations  characteristic  of  a  doubly-periodic  function  <I>  (z)  of 
the  third  kind  are 

=  <£(»,     <&(z  +  2a)')  =  e~  »~Z  Q(z), 


where  m  is  an  integer  different  from  zero. 

Obviously  the  number  of  zeros  in  a  parallelogram  is  a  constant,  as  well  as 
the  number  of  infinities.  Let  a  parallelogram,  chosen  so  that  its  sides 
contain  no  zero  and  no  infinity  of  <&  (z},  have  p,  p  +  2<w,  p  +  2&>'  for  three 
of  its  angular  points;  and  let  a1}  a2, . ..,  a{  be  the  zeros  and  cl5 ...,  cm  be  the 
infinities,  multiplicity  of  order  being  represented  by  repetitions.  Then  using 

"^  (z)  to  denote   ,    (log  <£  (z)},  we  have,   as   the    equations    characteristic   of 


* 


and  for  points  in  the  parallelogram 


where  -ff  (^)  has  no  infinity  within  the  parallelogram.     Hence 


the  integral  being  taken  round  the  parallelogram  :  by  using  the  Corollary  to 
Prop.  II.  in  §  116,  we  have 

27ri  (I  -  n)  -  -  \         -  \^L\  dz  = 
Jp  \  &>   / 

so  that  I  =  n  +  m : 

or  the  algebraical  excess  of  the  number  of  irreducible  zeros  over  the  number  of 

irreducible  infinities  is  equal  to  in. 

z 
Again,  since  —  =  1  + 


z  —  /A  z  —  p, 

a  c 

we  have  2 2 h  I  —  n  =  z"^  (z)  —  zH  (z), 

z  —  a         z  —  c 

and  therefore  2-Tn  (Sa  —  2c)  =  jz*\?  (z)  dz, 


142.]  OF   THE   THIRD    KIND  287 

the  integral  being  taken  round  the  parallelogram.     As  before,  this  gives 

rp+2<a'  rp+2<a  <  vnTri  "I 

2™  (2a  -  2c)  =  2ft)^  (z)  dz  -  MV  (z)  -       -  (z  +  2ft/)    dz. 

Jp  Jp  (  ft)  ) 

The  former  integral  is 

rp+*»'(g) 

,v  x  dz 
(*) 

miri 


for  the  side  of  the  parallelogram  contains*  no  zero  and  no  infinity 
The  latter  integral,  with  its  own  sign,  is 


<P(Z)  ft) 

=  0  +          {O  +  2«  +  2ft>')2  -  (p  +  2ft/)2} 

=  2TO7T*  (p  +  ft)  +  2ft)'). 

Hence  2a  —  Sc  =  m  (&)  +  2&/), 

giving  the  excess  of  the  sum  of  the  zeros  over  the  sum  of  the  infinities  in  any 
parallelogram  chosen  so  as  to  contain  the  variable  z  and  to  have  no  one  of  its 
sides  passing  through  a  zero  or  an  infinity  of  the  function. 

These  will  be  taken  as  the  irreducible  zeros  and  the  irreducible  infinities  : 
all  others  are  congruent  with  them. 

All  these  results  are  obtained  through  the  theorem  II.  of  §  116,  which 
assumes  that  the  argument  of  <y'  is  greater  than  the  argument  of  &)  or,  what 
is  the  equivalent  assumption  (§  129),  that 

rjco'  —  w'co  =  ^iri. 

143.  Taking  the  function,  naturally  suggested  for  the  present  class  by 
the  corresponding  function  for  the  former  class,  we  introduce  a  function 


a(z-  d)  <r(z-  C2).  ..<r(z  —  Cn)  ' 

where  the  a's  and  the  c's  are  connected  by  the  relations 
Sa  —  Sc  =  m  (&)  +  2&>'),     l—n  =  m. 

Then  (f>(z)  satisfies  the  equations  characteristic  of  doubly-periodic  functions 
of  the  third  kind,  if 

0  =  4Xo)  +  2ra77, 
k  .  27rt  =  4X&)2  +  2m?/ft)  +  2/ift)  +  miri  —  Zmrj  (&>  +  2ft)')  ; 


miri  —  2mrj'  (&>  +  2ft)'), 

*  Both  in  this  integral  and  in  the  next,  which  contain  parts  of  the  form   I  —    there  is,  as  in 

J    w 

Prop.  VII.,  §  116,  properly  an  additive  term  of  the  form  2iciri,  where  K  is  an  integer  ;  but,  as  there, 
both  terms  can  be  removed  by  modification  of  the  position  of  the  parallelogram,  and  this  modifi 
cation  is  supposed,  in  the  proof,  to  have  been  made. 


288  TERTIARY    FUNCTIONS  [143. 

k  and  k'  being  disposable  integers.     These  are  uniquely  satisfied  by  taking 


with  A;  =  0,     k'  =  m. 

Assuming  the  last  two,  the  values  of  X  and  /JL  are  thus  obtained  so  as  to  make 
<fr  (z)  a  doubly-periodic  function  of  the  third  kind. 

Now  let  Oj,  ...,  di  be  chosen  as  the  irreducible  zeros  of  <l>  (z)  and  Ci,  ...,  cn 
as  the  irreducible  infinities  of  <E>  (2),  which  is  possible  owing  to  the  conditions 
to  which  they  were  subjected.  Then  <3>  (z)/<j>  (z)  is  a  doubly-periodic  function 
of  the  first  kind;  it  has  no  zeros  and  no  infinities  in  the  parallelogram  of- 
periods  and  therefore  none  in  the  whole  plane  ;  it  is  therefore  a  constant,  so 
that 

3>  (z)  =  Ae"**  "IZ*+^  -  +  (l|+8'')}  **  <r(*-gi)°-(*-q»)-.  •*(*-<*!) 

tr(z-  d)  <r(z-  c.2)...o-  (z  -  cn)  ' 

a  representation  of  <3>  (z)  in  terms  of  known  quantities. 

Ex.     Had  the  representation  been  effected  by  means  of  the  Jacobian  Theta-Functions 
which  would  replace  a  (z)  by  H(z),  then  the  term  in  z1  in  the  exponential  would  be  absent. 

144.  No  limitation  on  the  integral  value  of  m,  except  that  it  must  not 
vanish,  has  been  made  :   and  the  form  just  obtained  holds  for  all  values. 
Equivalent  expressions  in  the  form  of  sums  of  functions  can  be  constructed  : 
but   there   is   then   a   difference   between   the  cases  of  m  positive  and  m 
negative. 

If  m  be  positive,  being  the  excess  of  the  number  of  irreducible  zeros  over 
the  number  of  irreducible  infinities,  the  function  is  said  to  be  of  positive  class 
m  ;  it  is  evident  that  there  are  suitable  functions  without  any  irreducible 
infinities  —  they  are  integral  functions. 

When  m  is  negative  (=  —  n),  the  function  is  said  to  be  of  negative  class  n  ; 
but  there  are  no  corresponding  integral  functions. 

145.  First,  let  m  be  positive. 

i.  If  the  function  have  no  accidental  singularities,  it  can  be  expressed  in 
the  form 

A  e**+i*  a-(z  —  a1)a-(z  —  aa)...<r(z  —  am), 

with  appropriate  values  of  X  and  //.. 

ii.  If  the  function  have  n  irreducible  accidental  singularities,  then  it  has 
m  +  n  irreducible  zeros.  We  proceed  to  shew  that  the  function  can  be 
expressed  by  means  of  similar  functions  of  positive  class  m,  with  a  single 
accidental  singularity. 


145.]  OF   POSITIVE   CLASS  289 

Using  X  and  /j,  to  denote 


,  mri 

-  1  —  '  and  |  -  -  +  m  (77  +  277'), 
&)  a) 

which  are  the  constants  in  the  exponential  factor  common  to  all  functions  of 
the  same  class,  consider  a  function,  of  positive  class  m  with  a  single  accidental 
singularity,  in  the  form 

*m  (z,  u)  =  eW  '' 


<r(u-  6X)  o-  (u  -  &„).  •  •  <r  (u  -  bm+1)  <r(z-u)' 
where  b1}  b.2, ...,  bm  are  arbitrary  constants,  of  sum  s,  and 
m  (&>  +  2ft)')  =  6OT+1  +  fcj  +  b.>  +  . . .  bm  -  u 

=  bm+l  +s-u. 
The  function  y-m  satisfies  the  equations 

_mirzi 

y-w  (z  +  2<w,  u)  =  i/rm  (z,  u),     y,tt  (z  +  2&)',  w)  =  e~'  «   -^m  (z,  u) ; 

regarded   as  a  function  of  z,  it  has  u   for  its  sole  accidental  singularity, 
evidently  simple. 

The  function  - — can  be  expressed  in  the  form 

I\I/*       I  £     It  I 

u  —  k) . . .  a-  (u  —  bm)         o-  {s  —  m  (&) 


(r^-b,)  ............  a-(z-bm)    a{u-  z-s  +  m(a>  +  2~w7)}  ' 

Regarded  as  a  function  of  u,  it  has  z,  \,  .  .  .,  bm  for  zeros  and  z  +  s  -  m  (to  +  2o>') 
for  its  sole  accidental  singularity,  evidently  simple  :  also 

z  +  &J  +  ...+  bm  -  {z  +  s  -  m  (&)  +  2o/)}  =  m  (w  +  2o>'). 

Hence  owing  to  the  values  of  X  arid  p,  it  follows  that   --  }  -  x    when  re- 

f>m(*,  tt) 

garded  as  a  function  of  u,  satisfies  all  the  conditions  that  establish  a  doubly- 
periodic  function  of  the  third  kind  of  positive  class  m,  so  that 

1  1 


~i 7 ~  =r  ^ 

and  therefore 


mnz 

tym  (z,  u  +  2o>)  =  ^m  (z,  u),     ^m  (z,  u  +  20)')  =  e~ijrm  (z,  u). 
Evidently  -f  m  (z,  u)  regarded  as  a  function  of  u  is  of  negative  class  m  :   its 
infinities  and  its  sole  zero  can  at  once  be  seen  from  the  form 

-bm)  o-{u-z-s+m(ca 


<r(u  -z)*^-^)...*^-  bm)  a-  {s  -  m  (to  +  2o)')j  ' 

Each  of  the  infinities  is  simple.     In  the  vicinity  of  u  =  z,  the  expansion  of 
the  function  is 

^^z  +  positive  integral  powers  of  u  —  z  : 

19 


290  TERTIARY  FUNCTIONS  [145. 

and,  in  the  vicinity  of  u  =  br,  it  is 

C*     (  7\ 

r  j    +  positive  integral  powers  of  u  —  br, 

Lv   "~~    \Jrp 

where  Gr  (z)  denotes 

r)  <r(z-bi)--.<r(z-br-i)<r(z-br+l)...a(z-bm)o-{z+s-br-m(a>+2a>')} 
a-  (br  -  6j).  ..cr  (br  -  6r_!)  cr(br  -  br+l)...cr(br  -  bin)  o-[s-ra(eo  +  2&>')}' 

and  is  therefore  an  integral  function  of  z  of  positive  class  m. 

Let  4>  (14)  be  a  doubly-periodic  function  of  the  third  kind,  of  positive  class 
m  ;  and  let  its  irreducible  accidental  singularities,  that  is,  those  which  occur 
in  a  parallelogram  containing  the  point  u,  be  a^  of  order  !+/*!,  a.,  of  order 
1  +  ju,2,  and  so  on.  In  the  immediate  vicinity  of  a  point  ar,  let 


--...  ± 


\ 

- 


rr—  r;r-...  r-,-~-     -       -rr. 

cm          du-  du^J  u  —  a,. 

Then  proceeding  as  in  the  case  of  the  secondary  doubly-periodic  functions 
(§  137),  we  construct  a  function 

F(u)  =  3?(u)^m(z,  u). 
We  at  once  have  F  (u  +  2o>)  =  F  (u)  =  F(u  +  2a>'), 

so  that  F(u)  is  a  doubly-periodic  function  of  the  first  kind;  hence  the  sum 
of  its  residues  for  all  the  poles  in  a  parallelogram  of  periods  is  zero. 

For  the  infinities  of  F  (u),  which  arise  through  the  factor  tym(z,  u},  wea 
have  as  the  residue  for  u  =  z 

-<*>(*), 
and  as  the  residue  for  u  =  br,  where  r  =  1,  2,  ...,  m, 


In  the  vicinity  of  a,.,  we  have 

fyn  (Z,  u)  =  ^rm  (Z,  «r)  +  (u  -  Or)  tym'  (z,  O.r) 


where  dashes  imply  differentiation  of  ^rm  {z,  u}  with  regard  to  u,  after  which 
u  is  made  equal  to  a,.  ;  so  that  in  <I>  (u)  tym  (z,  u)  the  residue  for  u  =  ar,  where 
r  =  l,  2,  ...,  is 

Er  (z)  =  Ar  ,jrm  (z,  ctr)  +  B,  Tjrm'  (z,  a,.)  +  Cr  tym"  (z,  ar)  +  ...+  Mr  <^m^r)  (z>  ar\ 
Hence  we  have 


and  therefore  ®(z)=  2  E,(z)+  2  <&  (br)  Gr(z), 

s=l  r=l 

giving  the  expression  of  <l>  (z)  by  means  of  doubly  -periodic  functions  of  tht 
third  kind,  which  are  of  positive  class  m  and  have  either  no  accidental  singu-> 
larity  or  only  one  and  that  a  simple  singularity. 


145.]  OF   NEGATIVE   CLASS  291 

The  m  quantities  blt  ...,  bm  are  arbitrary;  the  simplest  case  which  occurs 
is  when  the  m  zeros  of  &(z)  are  different  and  are  chosen  as  the  values 
of  &!,...,  bm.  The  value  of  3>(z)  is  then 

<&(*)=  2  JS'.C*), 

s=l 

where  the  summation  extends  to  all  the  irreducible  accidental  singularities  ; 
while,  if  there  be  the  further  simplification  that  all  the  accidental  singularities 
are  simple,  then 

<I>  (z)  =  A1  TJrm  (2,  «!>  +  As  tym  (z,  ot2)  +  .  .  ., 

the  summation  extending  to  all  the  irreducible  simple  singularities. 
The  quantity  tym  (z,  ar),  which  is  equal  to 

)    <r(z-bd...<r(z-  bm)  <r{z  +  2b-m(<o  +  2ft/)  -  ar] 


a-(ar  —  b1)...a-  (ar  -  bm)  <r  {26  -  m  (co  +  2ft>')}  a-  (z  -  ar)  ' 

and  is  subsidiary  to  the  construction  of  the  function    E  (z\   is   called  the 
simple  element  of  positive  class  m. 

In  the  general  case,  the  portion 


gives  an  integral  function  of  z,  and  the  portion  2  Es  (z)  gives  a  fractional 

s=l 

function  of  z. 

146.     Secondly,    let  m   be   negative    and   equal   to  —  n.      The    equations 
satisfied  by  &  (z}  are 


i  =  <I>  0),         <I>  (z  +  2ft)')  =  e  w   <£  0), 

and  the  number  of  irreducible  singularities  is  greater  by  n  than  the  number 
of  irreducible  zeros. 

One  expression  for  <i>  (z}  is  at  once  obtained  by  forming  its  reciprocal, 
which  satisfies  the  equations 

11  1  -2-**     i 

f\  /K   /  -\   > 


and  is  therefore  of  the  class  just  considered:    the  value  of is  of  the 

q>(^) 

form 

ZEs(z)  +  ^ArGr(z}. 

For  purposes  of  expansion,  however,  this  is  not  a  convenient  form  as  it  gives 
only  the  reciprocal  of  <I>  (z}. 

To  represent  the  function,  Appell  constructed  the  element 


TT    sv°°       Ffr-K»-*Wl        7r(2 
gr—    *  .    •  cot — *- 


19—2 


292  TERTIARY   FUNCTIONS  [146. 

which,  since  the  real  part  of  to' fan  is  positive,  converges  for  all  values  of  z  and 

y,  except  those  for  which 

z  =  y  (mod.  2&>,  2&>'). 

For   each  of  these  values  one  term  of  the  series,  and  therefore  the  series 
itself,  becomes  infinite  of  the  first  order. 

Evidently  %„  (z,  y  +  2o>)  =  %M  (z,  y}, 

niryi 

Xn (z,  y  +  2eo')  =  e    °    %„(*,  y); 
therefore  in  the  present  case 

0(y)=*(3f)jfr  (**?)> 

regarded  as  a  function  of  ^/,  is  a  doubly-periodic  function  of  the  first  kind. 

Hence  the  sum  of  the  residues  of  its  irreducible  accidental  singularities 
is  zero. 

When  the  parallelogram  is  chosen,  which  includes  z,  these  singularities 
are 

(i)     y  =  z,  arising  through  %n  (z,  y} ; 

(ii)    the  singularities  of  <£  (y},  which  are  at  least  n  in  number,  and  are 
n  +  I  when  <&  has  I  irreducible  zeros. 

The  expansion  of  Xn  0>  y),  in  powers  of  y  -  z,  in  the  vicinity  of  the  point 
z,  is 

+  positive  integral  powers  of  y  —  z  ; 


y-z 
therefore  the  residue  of  II  (y)  is 

Let  ctr  be  any  irreducible  singularity,  and  in  the  vicinity  of  a,,  let  3>  (y)  denote 

d 


-I- positive  integral  powers  of  y  —  Or, 

where   the   series   of  negative   powers   is  finite  because  the  singularity  is 
accidental ;   then  the  residue  of  H  (y}  is 

Ar  ^  (Z,  Or)  +  Br  Xn   (*,  «r)  +  Cr  %,/'  (z,  Ct,)  +  . . .  +  Pr  X*™  0>  «>')> 

where  %n(A)  (^,  ar)  is  the  value  of 

dx%n  (z,  y) 

dy* 

when  y  =  0r  after  differentiation.     Similarly  for  the  residues  of  other  singu 
larities  :  and  so,  as  their  sum  is  zero,  we  have 

<£  (Z)  =  2  {Ar  Xn  (*,  «r)  +  Br  Xn   (*,  «•,)  +  ...+  P,  XnW  (?,  «r)}, 

the  summation  extending  over  all  the  singularities. 


146.]  OF   NEGATIVE   CLASS  293 

The  simplest  case  occurs  when  all  the  N(>n)  singularities  a  are  accidental 
and  of  the  first  order  ;  the  function  4>  (z)  can  then  be  expressed  in  the  form 

Al  Xn  (Z,  «i)  +  A2  Xn  (Z,  Oj)  +  .  .  .  +  AN  Xn  (z,  «#)• 

The  quantity  Xn  (z,  a),  which  is  equal  to 

T    *^"     ^p{(«-i)»'+«}        TT  0  -  a 

a         2/6  COt  —  -^. 


. 

2(0 


is  called  the  simple  element  for  the  expression  of  a  doubly-periodic  function  of 
the  third  kind  of  negative  class  n. 


Ex.    Deduce  the  result 


_    ^    (  — iVcot 
TT   snu     s=-oov  I        2K        /' 

147.     The  function  Xn  (z,  y}  can  be  used  also  as  follows.     Since  Xm  (z,  y), 
qua  function  of  y,  satisfies  the  equations 

%m  (z,   11  +  2(i)}  =  Y™  (z,  7/\ 
llv    \      s      {/         '  /  /V//fc     \      J     ts  /' 

miryi 

Xm  (z,  y  +  2o/)  =  e~^xm  (z,  y), 

which  are  the  same  equations  as  are  satisfied  by  a  function  of  y  of  positive 
class  m,  therefore  Xm  (<*>  z),  which  is  equal  to 


2     e  cot 


being  a  function  of  z,  satisfies  the  characteristic  equations  of  §  142  ;  and,  in 
the  vicinity  of  z  =  a, 

Xm  (a>  z)  —  -   —  +  positive  integral  powers  of  z  —  a. 

Z  ~~"  OC 

If  then  we  take  the  function  4>  (z)  of  §  145,  in  the  case  when  it  has  simple 
singularities  at  alt  «2,  ...  and  is  of  positive  class  m,  then 


4>  (z)  +  A,  xw  (a,  , 

is  a  function  of  positive  class  m  without  any  singularities:  it  is  therefore 
equal  to  an  integral  function  of  positive  class  m,  say  to  G(z)t  where 

G  (z)  =  Ae^+^a-  (z-al}...(r(z-  am), 
so  that  3>(z)  =  G(z)-A1Xm(ct1,2)-A,xm(<Xt,z)-.... 

Ex.  As  a  single  example,  consider  a  function  of  negative  class  2,  and  let  it  have  no 
zero  within  the  parallelogram  of  reference.  Then  for  the  function,  in  the  canonical 
product-form  of  §  143,  the  two  irreducible  infinities  are  subject  to  the  relation 


and  the  function  is         *  (z)  =  AV°    "V"  - 

o-  (z—  Cj)  o-  (z-c2)' 


294  TERTIARY   FUNCTIONS  [147. 

The  simple  elements  to  express  3>  (z)  as  a  sum  are 

2.<!iri  ,        , 

»      {{s-lX  +  Cl}        ,77,  '    « 

*«  "  «rt       (s-C! -2*,), 


4iri,          ,, 
7T         -(ci-<o)»  -       r-w-c'i  TT 

=  _e<->  2  e  a>  cot  —  (2  +  0 j-2no) 


after  an  easy  reduction, 

4irj 


The  residue  of  *(s)  for  cn  which  is  a  simple  singularity,  is 

'Us-( 
Al  =  Ktfa       v< 

and  for  c2,  also  a  simple  singularity,  it  is 


, 

so  that  ^-  =  -ew  =-ew 

^2 

Hence  the  expression  for  4>  (z)  as  a  sum,  which  is 


! 

becomes  Al  (X2  (2,  Cj)  -  e  u     ^2  (^  -  ci)} 

that  is,  it  is  a  constant  multiple  of 


Again, 


—  j  -  - 

<r(z-  GJ)  a-  (z  +  c^  -  2o>- 


on  changing  the  constant  factor.     Hence  it  is  possible  to  determine  L  so  that 


•ni  Tti 

"  C'       «  c    -  e<a 


Taking  the  residues  of  the  two  sides  for  z=c1}  we  have 
and  therefore  finally  we  have 


-C]*-  —  Ci  --  C, 

Le™        <»     =  e    °> 


>-.•>-* 


TtlC 


(a  (s,  c)  -  e  w  X2  (2>  -  c) 


<*    Cot^L(2-c1-2su)')-e  w    cot  -  -  (z  +  cx  -  2sw') K 

2<a  2a>  ) 

the  right-hand  side  of  which  admits  of  further  modification  if  desired. 


147.]  PSEUDO-PERIODIC    FUNCTIONS  295 

Many  examples  of  such  developments  in  trigonometrical  series  are  given  by  Hermite*, 
Biehlerf,  HalphenJ,  Appell§,  and  Krause||. 

148.  We  shall  not  further  develop  the  theory  of  these  uniform  doubly- 
periodic  functions  of  the  third  kind.  It  will  be  found  in  the  memoirs  of 
Appell§  to  whom  it  is  largely  due;  and  in  the  treatises  of  Halphen**,  and 
of  Rausenberger"f"f. 

It  need  hardly  be  remarked  that  the  classes  of  uniform  functions  of  a 
single  variable  which  have  been  discussed  form  only  a  small  proportion  of 
functions  reproducing  themselves  save  as  to  a  factor  when  the  variable 
is  subjected  to  homographic  substitutions,  of  which  a  very  special  example 
is  furnished  by  linear  additive  periodicity.  Thus  there  are  the  various 
classes  of  pseudo-automorphic  functions,  (§  305)  called  Thetafuchsian  by  Pom- 
care,  their  characteristic  equation  being 


for  all  the  substitutions  of  the  group  determining  the  function  :  and  other 
classes  are  investigated  in  the  treatises  which  have  just  been  quoted. 

The  following  examples  relate  to  particular  classes  of  pseudo-periodic 
functions. 

Ex.  1.     Shew  that,  if  F  (z)  be  a  uniform  function  satisfying  the  equations 


m 

where  b  is  a  primitive  mth  root  of  unity,  then  F(z)  can  be  expressed  in  the  form 


where  f(z)  denotes  the  function 


and  prove  that  \F(z)dz  can  be  expressed  in   the  form   of  a  doubly-periodic   function 
together  with  a  sum  of  logarithms  of  doubly-periodic  functions  with  constant  coefficients. 

(Goursat.) 

*  Comptes  Rendus,  t.  Iv,  (1862),  pp.  11—18. 

t  Sur  les  developpements  en  series  des  fonctions  doublement  periodiqucs  de  troisieme  espece, 
(These,  Paris,  Gauthier-Villars,  1879). 

£  Traite  des  fonctions  elliptiques,  t.  i,  chap.  xm. 

§  Annales  de  VEc.  Norm.  Sup.,  3rae  S6r.,  t.  i,  pp.  135—164,  t.  ii,  pp.  9—36,  t.  iii,  pp.  9—42. 

||  Math.  Ann.,  t.  xxx,  (1887),  pp.  425—436,  516—534. 

'*  Traite  des  fonctions  elliptiques,  t.  i,  chap.  xiv. 

ft  Lehrbuch  der  Theorie  der  periodischen  Functional,  (Leipzig,  Teubner,  1884),  where  further 
references  are  given. 


296  PSEUDO-PERIODIC    FUNCTIONS  [148. 

Ex.  2.     Shew  that,  if  a  pseudo-periodic  function  be  denned  by  the  equations 


and  if,  in  the  parallelogram  of  periods  containing  the  point  z,  it  have  infinities  c,  ...  such 
that  in  their  immediate  vicinity 


then/  (2)  can  be  expressed  in  the  form 

-'^'^«{^I+  ......  +«»,£}«—>, 

the  summation  extending  over  all  the  infinities  of/  (z)  in  the  above  parallelogram  of  periods, 
and  the  constants  (715  ...  being  subject  to  the  condition 

+  iVS  Cl  =  A  o>'  —  X'«o. 

Deduce   an   expression   for  a   doubly-periodic   function  <f)  (z)  of  the   third   kind,    by 
assuming 

/W-f]8.  (Halphen.) 

(f>  \g) 

Ex.  3.     If    S(z)   be   a  given    doubly-periodic    function    of   the    first    kind,    then    a 
pseudo-periodic  function  F(z),  which  satisfies  the  equations 

F(z  +  ^}  =  F(z), 
mriz 
F  (z  +  2o>')  =  e  ~"~  S  (z}  F  (z), 

where  n  is  an  integer,  can  be  expressed  in  the  form 


where  -4  is  a  constant  and  TT  (2)  denotes 


the  summation  extending  over  all  points  &,.  and  the  constants  Br  being  subject  to  the 
relation 


Explain  how  the  constants  b,  G  and  B  can  be  determined.  (Picard.) 

Ex.  4.     Shew  that  the  function  F(z)  defined  by  the  equation 

for  values  of  \z\,  which  are  <1,  satisfies  the  equation 

and  that  the  function  Fl(a!)=^   ^rjr-£i 

where  (j)(,v)  =  3?  —  1,  and  </>„(.*•')>  f()r  positive  and  negative  values  of  n,  denotes  (/>  [0  {<£ 0  (#)}] 

<f>  being  repeated  n  times,  and  a  is  the  positive  root  of  a3  —  a  -  1  =  0 ;  satisfies  the  equation 

for  real  values  of  the  variable. 

Discuss  the  convergence  of  the  series  which  defines  the  function  Fl  (x).       (Appell.) 


CHAPTER  XIII. 

FUNCTIONS  POSSESSING  AN  ALGEBRAICAL  ADDITION-THEOREM. 

149.  WE  may  consider  at  this  stage  an  interesting  set*  of  important 
theorems,  due  to  Weierstrass,  which  are  a  justification,  if  any  be  necessary, 
for  the  attention  ordinarily  (and  naturally)  paid  to  functions  belonging  to 
the  three  simplest  classes  of  algebraic,  simply-periodic  and  doubly-periodic 
functions. 

A  function  <f>  (u)  is  said  to  possess  an  algebraical  addition  theorem,  when 
among  the  three  values  of  the  function  for  arguments  u,  v,  and  u  +  v,  where  u 
and  v  are  general  and  not  merely  special  arguments,  an  algebraical  equation 
exists  f  having  its  coefficients  independent  of  u  and  v. 

150.  It  is  easy  to  see,  from  one  or  two  examples,  that  the  function  does 
not   need   to   be   a  uniform  function  of  the  argument.     The  possibility  of 
multiformity  is  established  in  the  following  proposition : 

A  function  defined  by  an  algebraical  equation,  the  coefficients  of  which  are 
uniform  algebraical  functions  of  the  argument,  or  are  uniform  simply -periodic 
functions  of  the  argument,  or  are  uniform  doubly -periodic  functions  of  the 
argument,  possesses  an  algebraical  addition-theorem. 

*  They  are  placed  in  the  forefront  of  Schwarz's  account  of  Weierstrass's  theory  of  elliptic 
functions,  as  contained  in  the  Formeln  und  Lehrsdtze  zum  Gebrauche  der  elliptischen  Functionen; 
but  they  are  there  stated  (§§  1—3)  without  proof.  The  only  proof  that  has  appeared  is  in  a 
memoir  by  Phragmen,  Acta  Math.,  t.  vii,  (1885),  pp.  33—42;  and  there  are  some  statements 
(pp.  390—393)  in  Biermann's  Theorie  der  analytischen  Functionen  relative  to  the  theorems.  The 
proof  adopted  in  the  text  does  not  coincide  with  that  given  by  Phragme'n. 

t  There  are  functions  which  possess  a  kind  of  algebraical  addition -theorem ;  thus,  for 
instance,  the  Jacobian  Theta-functions  are  such  that  eA(u  +  w)  O^  (u-  v)  can  be  rationally  ex 
pressed  in  terms  of  the  Theta-functions  having  it  and  v  for  their  arguments.  Such  functions 
are,  however,  naturally  excluded  from  the  class  of  functions  indicated  in  the  definition. 

Such  functions,  however,  possess  what  may  be  called  a  multiplication-theorem  for  multipli 
cation  of  the  argument  by  an  integer,  that  is,  the  set  of  functions  6  (nut)  can  be  expressed 
algebraically  in  terms  of  the  set  of  functions  6  (M).  This  is  an  extremely  special  case  of  a  set 
of  transcendental  functions  having  a  multiplication-theorem,  which  are  investigated  by  Poincare, 
Liouville,  4°"  S6r.,  t.  iv,  (1890),  pp.  313—365. 


298  EXAMPLES   OF   FUNCTIONS  [150. 

First,  let  the  coefficients  be  algebraical  functions  of  the  argument  u.  If 
the  function  defined  by  the  equation  be  U,  we  have 

Umg0  (u)  +  Um~lgi  (u)  +  ...+gm  (u)  =  0, 

where  g0(u),gi(u},  ...,gm(u)  are  rational  integral  algebraical  functions  of  u 
of  degree,  say,  not  higher  than  n.  The  equation  can  be  transformed  into 

un  f/U\+  u'1-1/!  (  U)  +  ...  +  fn  (  U)  =  0, 

where  f0(U),  fi(U),  ••••>  fn(U)  are  rational  integral  algebraical  functions  of 
U  of  degree  not  higher  than  m. 

If  V  denote  the  function  when  the  argument  is  v,  and  W  denote  it  when 
the  argument  is  u  +  v,  then 

w»/0  (7)  +  ^1/1  (7)  +  ...  +fn  (V)  M  0, 
and  (u  +  v)n/0  (  W)  +  (u  +  vY^f,  (  W )  +  . . .  +fn  ( W )  =  0. 

The  algebraical  elimination  of  the  two  quantities  u  and  v  between  these 
three  equations  leads  to  an  algebraical  equation  between  the  quantities 
/(£/"),  /(7)  and  f  (W),  that  is,  to  an  algebraical  equation  between  U,  V,  W, 
say  of  the  form 

G(U,  V,  F)  =  0, 

where  G  denotes  an  algebraical  function,  with  coefficients  independent  of 
u  and  v.  It  is  easy  to  prove  that  G  is  symmetrical  in  U  and  7,  and  that 
its  degree  in  each  of  the  three  quantities  U,  7,  W  is  wn2.  The  equation 
G  =  0  implies  that  the  function  U  possesses  an  algebraical  addition- theorem. 

Secondly,  let  the  coefficients*  be  uniform  simply-periodic  functions  of 
the  argument  u.  Let  &>  denote  the  period:  then,  by  §  113,  each  of  these 

TT'IL 

functions    is   a    rational    algebraical    function    of    tan  — .     Let   u'    denote 

tan  — ;   then  the  equation  is  of  the  form 

Umg0  (u')  +  Um^g,  (u'}  +  ...+  gm  00  =  0, 

where  the  coefficients  g  are  rational  algebraical  (and  can  be  taken  as 
integral)  functions  of  u'.  If  p  be  the  highest  degree  of  u'  in  any  of  them, 
then  the  equation  can  be  transformed  into 

u'vfo  (  U)  +  u'P-1/!  (  U)  +  . . .  +  fp  ( U)  =  0, 

where  f0(U),  fi(U),  ...,  fp(U)  are  rational  integral  algebraical  functions  of 
U  of  degree  not  higher  than  m. 

*  The  limitation  to  uniformity  for  the  coefficients  has  been  introduced  merely  to  make  the 
illustration  simpler;  if  in  any  case  they  were  multiform,  the  equation  would  be  replaced  by 
another  which  is  equivalent  to  all  possible  forms  of  the  first  arising  through  the  (finite) 
multiformity  of  the  coefficients :  and  the  new  equation  would  conform  to  the  specified 
conditions. 


150.]  POSSESSING    AN   ADDITION-THEOREM  299 

Let  v  denote  tan  —  ,  and  w  denote  tan  —  --  ;  then  the  corresponding 
cy  &) 

values  of  the  function  are  determined  by  the  equations 


and  w'*>f0(W)  +  w'p-*/!  (W)  +  ...  +fp  (W)  =  0. 

The  relation  between  u',  v',  w'  is 

u'v'w'  +  u'  +  v'  -  w'  =  0. 

The  elimination  of  the  three  quantities  u',  v',  w'  among  the  four  equations 
leads  as  before  to  an  algebraical  equation 

G(U,  V,  W)  =  0, 

where  G  denotes  an  algebraical  function  (now  of  degree  mp'2)  with  coefficients 
independent  of  u  and  v.  The  function  U  therefore  possesses  an  algebraical 
addition-theorem. 

Thirdly,  let  the  coefficients  be  uniform  doubly-periodic  functions  of  the 
argument  u.  Let  &>  and  &/  be  the  two  periods  ;  and  let  @  (u),  the  Weier- 
strassian  elliptic  function  in  those  periods,  be  denoted  by  £.  Then  every 
coefficient  can  be  expressed  in  the  form 


~L          ' 

where  L,  M,  N  are  rational  integral  algebraical  functions  of  f  of  finite 
degree.  Unless  each  of  the  quantities  N  is  zero,  the  form  of  the  equation 
when  these  values  are  substituted  for  the  coefficients  is 

A+Bp'(u)  =  0, 

so  that  A*  =  &(±?-g£-9*)\ 

and  this  is  of  the  form 

Umff*  (£)  +  U'^g,  (|)  +  .  .  .  +  gm  (£)  -  0, 

where  the  coefficients  g  are  rational  algebraical  (and  can  be  taken  as  integral) 
functions  of  £  If  q  be  the  highest  degree  of  £  in  any  of  them,  the  equation 
can  be  transformed  into 


where  the  coefficients  /  are  rational  integral  algebraical  functions  of  U  of 
degree  not  higher  than  2m. 

Let  TJ  denote  $  (v)  and  f  denote  p(u  +  v);  then  the  corresponding  values 
of  the  function  are  determined  by  the  equations 

.........  +fq(V)=0, 


By  using  Ex.  4,  §  131,  it  is  easy  to  shew  that  the  relation  between  £,  rj,  £  is 


300  WEIERSTRASS'S   THEOREM    ON    FUNCTIONS  [150. 

The  elimination  of  £,  ij,  £  from  the  three  equations  leads  as  before  to  an 

algebraical  equation 

G(U,V,  W)  =  0, 

of  finite  degree  and  with  coefficients  independent  of  u  and  v.    Therefore  in  this 
case  also  the  function  U  possesses  an  algebraical  addition-theorem. 

If,  however,  all  the  quantities  N  be  zero,  the  equation  defining  U  is  of  the 

form 

Umh0  (£)  +  U^h,  (£)  +  . . .  +  hm  (£)  =  0, 

and  a  similar  argument  then  leads  to  the  inference  that    U  possesses   an 
algebraical  addition-theorem. 

The  proposition  is  thus  completely  established. 

151.  The  generalised  converse  of  the  preceding  proposition  now  suggests 
itself :  what  are  the  classes  of  functions  of  one  variable  that  possess  an  alge 
braical  addition-theorem?  The  solution  is  contained  in  Weierstrass's  theorem : — 

An  analytical  function  <f>  (u),  which  possesses  an  algebraical  theorem,  is 
either 

(i)   an  algebraical  function  of  u  ;  or 

liru 

(ii)  an   algebraical  function   of   e  »  ,   where   w    is    a    suitably   chosen 
constant ;  or 

(iii)  an  algebraical  function  of  the  elliptic  function  %>(u),  the  periods — or 
the  invariants  g.z  and  g3 — being  suitably  chosen  constants. 

Let  U  denote  </>  (w). 

For  a  given  general  value  of  u,  the  function  U  may  have  m  values  where, 
for  functions  in  general,  there  is  not  a  necessary  limit  to  the  value  of  m ;  it 
will  be  proved  that,  when  the  function  possesses  an  algebraical  addition- 
theorem,  the  integer  m  must  be  finite. 

For  a  given  general  value  of  U,  that  is,  a  value  of  U  when  its  argument  is 
not  in  the  immediate  vicinity  of  a  branch-point  if  there  be  branch-points,  the 
variable  u  may  have  p  values,  where  p  may  be  finite  or  may  be  infinite. 

Similarly  for  given  general  values  of  v  and  of  V,  which  will  be  used  to 
denote  <£  (v). 

First,  let  p  be  finite.  Then  because  u  has  p  values  for  a  given  value  of  U 
and  v  has  p  values  for  a  given  value  of  V,  and  since  neither  set  is  affected  by  the 
value  of  the  other  function,  the  sum  u  +  v  has  p2  values  because  any  member  of 
the  set  u  can  be  combined  with  any  member  of  the  set  v  ;  and  this  number 
p2  of  values  of  u  +  v  is  derived  for  a  given  value  of  U  and  a  given  value  of  V. 

Now  in  forming  the  function  <j>(u  +  v),  which  will  be  denoted  by  W,  we 
have  m  values  of  W  for  each  value  of  u  +  v  and  therefore  we  have  mp2  values 
of  W  for  the  whole  set,  that  is,  for  a  given  value  of  U  and  a  given  value  of  V. 


151.]  POSSESSING   AN   ADDITION-THEOREM  301 

Hence  the  equation  between   U,  V,  W  is  of  degree*  mp2  in  W,  necessarily 
finite  when  the  equation  is  algebraical  ;  and  therefore  m  is  finite. 

Because  m  is  finite,  U  has  a  finite  number  m  of  values  for  a  given  value  of 
u  ;  and,  because  p  is  finite,  u  has  a  finite  number  p  of  values  for  a  given  value  of 
U.  Hence  U  is  determined  in  terms  of  u  by  an  algebraical  equation  of  degree 
m,  the  coefficients  of  which,  are  rational  integral  algebraical  functions  of 
degree  p  ;  and  therefore  U  is  an  algebraic  function  of  u. 

152.  Next,  let  p  be  infinite  ;  then  (see  Note,  p.  303)  the  system  of  values 
may  be  composed  of  (i)  a  single  simply-infinite  series  of  values  or  (ii)  a  finite 
number  of  simply-infinite  series  of  values  or  (iii)  a  simply-infinite  number  of 
simply-infinite  series  of  values,  say,  a  single  doubly-infinite  series  of  values  or 
(iv)  a  finite  number  of  doubly-infinite  series  of  values  or  (v)  an  infinite 
number  of  doubly-infinite  series  of  values  where,  in  (v),  the  infinite  number 
is  not  restricted  to  be  simply-infinite. 

Taking  these  alternatives  in  order,  we  first  consider  the  case  where  the  p 
values  of  u  for  a  given  general  value  of  U  constitute  a  single  simply  -infinite 
series.  They  may  be  denoted  by  f  (u,  n),  where  n  has  a  simply-infinite 
series  of  values  and  the  form  of/  is  such  that  f(u,  0)  =  u. 

Similarly,  the  p  values  of  v  for  a  given  general  value  of  V  may  be  denoted 
by/(y,  n),  where  n'  has  a  simply-infinite  series  of  values.  Then  the  different 
values  of  the  argument  for  the  function  W  are  the  set  of  values  given  by 

f(u,n)+f(v,ri), 

for  the  simply-infinite  series  of  values  for  n  and  the  similar  series  of  values 
for  n'. 

The  values  thus  obtained  as  arguments  of  W  must  all  be  contained  in 
the  series  f(u  +  v,  n"},  where  n"  has  a  simply-infinite  series  of  values  ;  and, 
in  the  present  case,/(w  +  w,  n"}  cannot  contain  other  values.  Hence  for  some 
values  of  n  and  some  values  of  n',  the  total  aggregate  being  not  finite,  the 
equation 

f(u,n}+f(v,n'}=f(u  +  v,n") 
must  hold,  for  continuously  varying  values  of  u  and  v. 

In  the  first  place,  an  interchange  of  u  and  v  is  equivalent  to  an  interchange 
of  n  and  n  on  the  left-hand  side;  hence  n"  is  symmetrical  in  n  and  n'. 
Again,  we  have 

df(u,  n)  _  df(u  +  v,  n") 
du  3  (u  +  v) 


dv      ' 

*  The  degree  for  special  functions  may  be  reduced,  as  in  Cor.  1,  Prop.  XIII,  §  118;  but  in  no 
case  is  it  increased.  Similarly  modifications,  in  the  way  of  finite  reductions,  may  occur  in  the 
succeeding  cases  ;  but  they  will  not  be  noticed,  as  they  do  not  give  rise  to  essential  modification 
in  the  reasoning. 


302  FORM   OF   ARGUMENT  [152. 

so  that  the  form  of  f(u,  n)  is  such  that  its  first  derivative  with  regard  to  u  is 
independent  of  u.  Let  0  (n)  be  this  value,  where  0  (n),  independent  of  u,  may 
be  dependent  on  n  ;  then,  since 


we  have  f(u,  n)  =  uO  (n)  +  ty  (n), 

-fy-  (n)  being  independent  of  u.     Substituting  this  expression  in  the  former 

equation,  we  have  the  equation 

u6  (n)  +  ^  (n)  +  v9  (n'}  +  f  (71')  =  (u  +  v)6  (n"}  +  ^  (n"), 
which  must  be  true  for  all  values  of  u  and  v  ;  hence 

e(n)=e(n")  =  d(n'), 

so  that  6  (n)  is  a  constant  and  equal  to  its  value  when  n  =  0.     But  when  n  is 
zero,/(w,  0)  is  u  ;  so  that  9  (0)  =  1  and  ^  (0)  =  0,  and  therefore 

f(u,  n)  =  u  +  Tjr  (n), 
where  i/r  vanishes  with  n. 

The  equation  defining  ty  is 


for  values  of  n  from  a  singly-infinite  series  and  for  values  of  n'  from  the  same 
series,  that  series  is  reproduced  for  TO".     Since  ^  (n)  vanishes  with  n,  we  take 

^  (n)  =  HX  (n), 

and  therefore  rc%  (n)  +  n'%  (n')  =  ri'x  (n"). 

Again,  when  n'  vanishes,  the  required  series  of  values  of  n"  is  given  by  taking 
n"  =  n  ;  and,  when  n   does  not  vanish,  n"  is  symmetrical  in  n  and  n',  so  that 

we  have 

n"  =  n  +  n'  +  nn\, 

where  X  is  not  infinite  for  zero  or  finite  values  of  n  or  n'.     Thus 
•HX  (n)  +  n'x  (n)  =  (n  +  TO'  +  -nw'X)  %  (w  +  ?*'  +  wi'X). 

Since  the  left-hand  side  is  the  sum  of  two  functions  of  distinct  and  inde 
pendent  magnitudes,  the  form  of  the  equation  shews  that  it  can  be  satisfied 

only  if 

X  =  0,  so  that  n"  =  n  +  n'  ; 

and  %  0)  =  %  (n//) 

=  %(n'\ 
so  that  each  is  a  constant,  say  o>  ;  then 

f(u,  n}  =  u  +  nco, 

which  is  the  form  that  the  series  must  adopt  when  the  series  f(u  +  v,  n")  is 
obtained  by  the  addition  of/(«,  n)  and/0,  n')- 


152.]  IN   A   SIMPLY-INFINITE    SERIES  303 

It  follows  at  once  that  the  single  series  of  arguments  for  W  is  obtained, 
as  one  simply-infinite  series,  of  the  form  u  +  v+n"a).  For  each  of  these 
arguments  we  have  m  values  of  W,  and  the  set  of  m  values  of  W  is 
the  same  for  all  the  different  arguments;  that  is,  W  has  m  values  for  a 
given  value  of  U  and  a  given  value  of  V.  Moreover,  U  has  m  values  for  each 
argument  and  likewise  V;  hence,  as  the  equation  between  U,  V,  W  is  of 
a  degree  that  is  necessarily  finite  because  the  equation  is  algebraical,  the 
integer  m  is  finite. 

It  thus  appears  that  the  function  U  has  a  finite  number  m  of  values  for 
each  value  of  the  argument  u,  and  that  for  a  given  value  of  the  function  the 
values  of  the  argument  form  a  simply-periodic  series  represented  by  u  +  nw. 

But  the  function  tan  (  —  )  is  such  that,  for  a  given  value,  the  values  of  the 

V  03  J 

argument  are  represented  by  the  series  u  +  nw  ;    hence  for  each  value  of 

tan  (  —  1  there  are  m  values  of  U  and  for  each  value  of  U  there  is  one  value 
\  «o  / 

of  tan  --  .    It  therefore  follows,  by  SS  113,  114,  that  between  U  and  tan  (—  } 
w  \  to  / 

there  is  an  algebraical  relation  which  is  of  the  first  degree  in  tan  -  -  and  the 


O) 

U 


rath  degree  in  U,  that  is,  U  is  an  algebraic  function  of  tan  —  -  .     Hence  U  is 


(I) 


an  algebraic  function  also  of  e  <"  . 

Note.  This  result  is  based  upon  the  supposition  that  the  series  of  argu 
ments,  for  which  a  branch  of  the  function  has  the  same  value,  can  be  arranged 
in  the  form/(w,  n),  where  n  has  a  simply-infinite  series  of  integral  values.  If, 
however,  there  were  no  possible  law  of  this  kind — the  foregoing  proof  shews 
that,  if  there  be  one  such  law,  there  is  only  one  such  law,  with  a  properly 
determined  constant  co — then  the  values  would  be  represented  by  ul}  u»,  ...,up 
with  p  infinite  in  the  limit.  In  that  case,  there  would  be  an  infinite  number  of 
sets  of  values  for  u  +  v  of  the  type  WA  +  v^,  where  X  and  p  might  be  the  same 
or  might  be  different ;  each  set  would  give  a  branch  of  the  function  W  and  then 
there  would  be  an  infinite  number  of  values  of  W  corresponding  to  one  branch 
of  U  and  one  branch  of  V.  The  equation  between  U,  V  and  W  would  be  of 
infinite  degree  in  W,  that  is,  it  would  be  transcendental  and  not  algebraical. 
The  case  is  excluded  by  the  hypothesis  that  the  addition-theorem  is  alge 
braical,  and  therefore  the  equation  between  U,  V  and  W  is  algebraical. 

153.  Next,  let  there  be  a  number  of  simply-infinite  series  of  values  of 
the  argument  of  the  function,  say  q,  where  q  is  greater  than  unity  and 
may  be  either  finite  or  infinite.  Let  ul}  u.2,  ...,  uq  denote  typical  members 
of  each  series. 

Then  all  the  members  of  the  series  containing  ul  must  be  of  the  form 


304  FORM   OF   ARGUMENT  [153. 

fi  (ui>  n)>  f°r  an  infinite  series  of  values  of  the  integer  n.  Otherwise,  as  in  the 
preceding  note,  the  sum  of  the  values  in  the  series  of  arguments  u  and  of 
those  in  the  same  series  of  arguments  v  would  lead  to  an  infinite  number  of 
distinct  series  of  values  of  the  argument  u  +  v,  with  a  corresponding  infinite 
number  of  values  W  ;  and  the  relation  between  U,  V,  W  would  cease  to  be 
algebraical. 

In  the  same  way,  the  members  of  the  corresponding  series  containing  ^ 
must  be  of  the  form/!  (v1}  ri)  for  an  infinite  series  of  values  of  the  integer  n'. 
Among  the  combinations 


the  simply-infinite  series  fi(tii+v1}  n")  must  occur  for  an  infinite  series 
of  values  of  n";  and  therefore,  as  in  the  preceding  case, 

fi(uly  n)  =  M1  +  nw1, 

where  toj  is  an  appropriate  constant.  Further,  there  is  only  one  series  of 
values  for  the  combination  of  these  two  series  ;  it  is  represented  by 

Ui  +  v1  +  n"wl. 

In  the  same  way,  the  members  of  the  series  containing  u2  can  be  repre 
sented  in  the  form  u2  +  nco2,  where  o>2  is  an  appropriate  constant,  which  may 
be  (but  is  not  necessarily)  the  same  as  Wj  ;  and  the  series  containing  u.2, 
when  combined  with  the  set  containing  v2,  leads  to  only  a  single  series 
represented  in  the  form  u.2  +  v2  +  ri'o)2.  And  so  on,  for  all  the  series  in  order. 

But  now  since  u2  +  m2a)2,  where  m2  is  an  integer,  is  a  value  of  u  for  a  given 
value  of  U,  it  follows  that  U  (u2  +  ra2a>2)  =  U  (w2)  identically,  each  being  equal 

to  U.     Hence 

U  (M!  +  mlwl  +  7n.,<y2)  =  U  (i^  +  ra^)  =  U  (u^  =  U, 

and  therefore  ^  +  ml(al  +  ra2&>2  is  also  a  value  of  u  for  the  given  value  of  U, 
leading  to  a  series  of  arguments  which  must  be  included  among  the  original 
series  or  be  distributed  through  them.  Similarly  u1  +  2mr(i)r,  where  the 
coefficients  ra  are  integers  and  the  constants  to  are  properly  determined, 
represents  a  series  of  values  of  the  variable  u,  included  among  the  original 
series  or  distributed  through  them.  And  generally,  when  account  is  taken  of 
all  the  distinct  series  thus  obtained,  the  aggregate  of  values  of  the  variable  u 
can  be  represented  in  the  form  Wx+2wrtur,  for  \  —  1,  2,  ...,  K,  where  K  is 
some  finite  or  infinite  integer. 

Three  cases  arise,  (a)  when  the  quantities  «  are  equal  to  one  another  or 
can  be  expressed  as  integral  multiples  of  only  one  quantity  a>,  (6)  when  the 
quantities  &>  are  equivalent  to  two  quantities  f^  and  O2  (the  ratio  of  which  is 
not  real),  so  that  each  quantity  &>  can  be  expressed  in  the  form 

a>r=plrfil+parsia> 

the  coefficients  plr,  p2r  being  finite  integers  ;  (c)  when  the  quantities  «  are 
not  equivalent  to  only  two  quantities,  such  as  flj  and  fl2. 


153.]  SIMPLY-PERIODIC   FUNCTIONS  305 

For  case  (a),  each  of  the  K  infinite  series  of  values  u  can  be  expressed 
in  the  form  u^+pci),  for  X  =  1,  2,  ...,  «  and  integral  values  of  p. 

First,  let  K  be  finite,  so  that  the  original  integer  q  is  finite.  Then  the 
values  of  the  argument  for  W  are  of  the  type 


that  is,  MA  +  '?V  +£>"&>, 

for  all  combinations  of  \  and  fju  and  for  integral  values  of  p".  There  are  thus 
K-  series  of  values,  each  series  containing  a  simply-infinite  number  of  terms 
of  this  type. 

For  each  of  the  arguments  in  any  one  of  these  infinite  series,  W  has  ra 
values  ;  and  the  set  of  m  values  is  the  same  for  all  the  arguments  in  one  and 
the  same  infinite  series.  Hence  W  has  w/c2  values  for  all  the  arguments  in 
all  the  series  taken  together,  that  is,  for  a  given  value  of  U  and  a  given 
value  of  V.  The  relation  between  U,  V,  W  is  therefore  of  degree  m«2, 
necessarily  finite  when  the  equation  is  algebraical  ;  hence  m  is  finite. 

It  thus  appears  that  the  function  U  has  a  finite  number  m  of  values  for 
each  value  of  the  argument  u,  and  that  for  a  given  value  of  the  function  there 
are  a  finite  number  K  of  distinct  series  of  values  of  the  argument  of  the  form 

7TU 

u+poi),  w  being  the  same  for  all  the  series.  But  the  function  tan  --  has 
one  value  for  each  value  of  u  and  the  series  u+pat  represents  the  series  of 

7TU 

values  of  u  for  a  given  value  of  tan  —  .     It  therefore  follows  that  there  are 

CO 

m  values  of  U  for  each  value  of  tan  —  and  that  there  are  K  values  of  tan  — 

to  o> 

for  each  value  of  U  ;  and  therefore  there  is  an  algebraical  relation  between 

U  and  tan  —  ,  which  is  of  degree  K  in  the  latter  and  of  degree  m  in  the 
&) 

iiru 
TTlI 

former.    Hence  U  is  an  algebraic  function  of  tan  —  and  therefore  also  of  e  M  . 


Next,  let  K  be  infinite,  so  that  the  original  integer  q  is  infinite.  Then, 
as  in  the  Note  in  §  152,  the  equation  between  U,  V,  W  will  cease  to  be 
algebraical  unless  each  aggregate  of  values  u^+pw,  for  each  particular 
value  of  p  and  for  the  infinite  sequence  X=  1,  2,  ...,  K,  can  be  arranged  in  a 
system  or  a  set  of  systems,  say  a  in  number,  each  of  the  form  fp(u+pa),  pp) 
for  an  infinite  series  of  values  of  pp.  Each  of  these  implies  a  series  of  values 
fp(v+p'u>,  pp)  of  the  argument  of  V  for  the  same  series  of  values  of  pp  as  of 
pp>  and  also  a  series  of  values  fp(u  +  v+p"(o,  pp")  of  the  argument  of  W  for 
the  same  series  of  values  of  pp".  By  proceeding  as  in  §  152,  it  follows  that 

fp  (u  +pa>,  pp}  =  u+pto  +pp(0p, 

where  &>p'  is  an  appropriate  constant,  the  ratio  of  which  to  &>  can  be  proved 
F.  20 


306  FORM   OF   ARGUMENT  [153. 

(as  in  §  106)  to  be  not  purely  real,  and  pp  has  a  simply-infinite  succession  of 
values.     The  integer  a  may  be  finite  or  it  may  be  infinite. 

When  ay  and  all  the  constants  o>'  which  thus  arise  are  linearly  equivalent 
to  two  quantities  f^  and  O2,  so  that  the  terms  additive  to  u  can  be  expressed 
in  the  form  8^  +  s.2fl»,  then  the  aggregate  of  values  u  can  be  expressed 
in  the  form 


for  a  simply-infinite  series  for  pl  and  for  p2  ;  and  p  has  a  series  of  values 
1,  2,  ...,  <r.  This  case  is,  in  effect,  the  same  as  case  (6). 

When  o)  and  all  the  constants  «'  are  not  linearly  equivalent  to  only 
two  quantities,  such  as  Oj  and  IL>,  we  have  a  case  which,  in  effect,  is  the 
same  as  case  (c). 

These  two  cases  must  therefore  now  be  considered. 

For  case  (6),  either  as  originally  obtained  or  as  derived  through  parfc 
of  case  (a),  each  of  the  (doubly)  infinite  series  of  values  of  u  can  be  expressed 
in  the  form 


for  X  =  1,  2,  ...,  <r  and  for  integral  values  of  _p,  and  p,.     The  integer  a  may  be 
finite  or  infinite  ;  the  original  integer  q  is  infinite. 

First,  let  cr  be  finite.     Then  the  values  of  the  argument  for  W  are  of  the 
type 


that  is,  u\  +  v^  +pi"£li  +  p2"O2, 

for  all  combinations  of  \  and  p  and  for  integral  values  of  £>/'  and  p.".     There 
are  thus  cr2  series  of  values,  each  series  containing  a  doubly-infinite  number  ofl 
terms  of  this  type. 

For  every  argument  there  are  m  values  of  W  ;  and  the  set  of  m  values  is 
the  same  for  all  the  arguments  in  one  and  the  same  infinite  series.     Thus  W 
has  mo-2  values  for  all  the  arguments  in  all  the  series,  that  is,  for  a  given  value 
of  U  and  a  given  value  of  V;  and  it  follows,  as  before,  from  the  consideration  i 
of  the  algebraical  relation,  that  m  is  finite. 

The  function  U  thus  has  m  values  for  each  value  of  the  argument  u  ;  and 
for  a  given  value  of  the  function  there  are  cr  series  of  values  of  the  argument, 
each  series  being  of  the  form  wx  +  PI^I  +p.2Q*- 

Take  a  doubly-periodic  function  ©  having  Oj  and  H2  for  its  periods,  such*1 
that  for  a  given  value  of  ©  the  values  of  its  arguments  are  of  the  foregoing 
form.     Whatever  be  the  expression  of  the  function,  it  is  of  the  order  cr.  , 
Then   U  has  m  values  for  each  value  of  @,  and  @  has  one  value  for  each'. 
value  of  U;  hence  there  is  an  algebraical  equation  between   U  and  ©,  ow 

*  All  that  is  necessary  for  this  purpose  is  to  construct,  by  the  use  of  Prop.  XII,  §  118,  ai 
function  having,   as  its  irreducible  simple  infinities,  a  series  of  points  aj,  a2,...,  a<7  —  special* 
values  of  «j,  w2,  ...,  ua—  in  the  parallelogram  of  periods,  chosen  so  that  no  two  of  the  <r  points  a 
coincide. 


153.]  DOUBLY-PERIODIC    FUNCTIONS  307 

:he  first  degree  in  the  latter  and  of  the  rath  degree  in  U:  that  is,  U  is  an 
algebraical  function  of  @.  But,  by  Prop.  XV.  §  119,  ©  can  be  expressed  in 
the  form 


where  L,  M,  N  are  rational  integral  algebraical  functions  of  $  (u),  if  f^  and  H2 
be  the  periods  of  g)  (u);  and  g)'  (u)  is  a  two-  valued  algebraical  function  of  jjp  (u), 
so  that  ©  is  an  algebraical  function  of  i@  (u).     Hence  also  U  is  an  algebraical 
function  of  $(u\  the  periods  o/<p  (u)  being  properly  chosen. 

This  inference  requires  that  a,  the  order  of  ©,  be  greater  than  1. 
Because  U  has  m  values  for  an  argument  u,  the  symmetric  function  St/" 
has  one  value  for  an  argument  u  and  it  is  therefore  a  uniform  function. 
But  each  term  of  the  sum  has  the  same  value  for  u+pifli+pflt  as  for 
u  ;  and  therefore  this  uniform  function  is  doubly-periodic.  The  number  of 
independent  doubly-infinite  series  of  values  of  u  for  a  uniform  doubly- 
periodic  function  is  at  least  two  :  and  therefore  there  must  be  at  least  two 
doubly-infinite  series  of  values  of  u,  so  that  <r  >  1.  Hence  a  function,  that 
possesses  an  addition-theorem,  cannot  have  only  one  doubly-infinite  series  of 
values  for  its  argument. 

If  cr  be  infinite,  there  is  an  infinite  series  of  values  of  u  of  the  form 
+  p^  +  p.flz  ;  an  argument,  similar  to  that  in  case  (a),  shews  that  this  is, 
in  effect,  the  same  as  case  (c). 

It  is  obvious  that  cases  (ii),  (iii)  and  (iv)  of  §  152  are  now  completely 
covered  ;  case  (v)  of  §  152  is  covered  by  case  (c)  now  to  be  discussed  in  §  154. 

154.  For  case  (c),  we  have  the  series  of  values  u  represented  by  a  number 
of  series  of  the  form 


where  the  quantities  &>  are  not  linearly  equivalent  to  two  quantities  flj  and 
Q2-     The  original  integer  q  is  infinite. 

Then,  by  §§  108,  110,  it  follows  that  integers  m  can  be  chosen  in  an 
unlimited  variety  of  ways  so  that  the  modulus  of 


r=l 

is  infinitesimal,  and  therefore  in  the  immediate  vicinity  of  any  point  u^ 
there  is  an  infinitude  of  points  at  which  the  function  resumes  its  value. 
Such  a  function  would,  as  in  previous  instances,  degenerate  into  a  mere 
constant  ;  and  therefore  the  combination  of  values  which  gives  rise  to  this 
case  does  not  occur. 

All  the  possible  cases  have  been  considered:  and  the  truth  of  Weierstrass's 

20—2 


308  EXAMPLES  [154. 

theorem*  that  a  function,  which  has  an  algebraical  addition-theorem,  is  either 

imi 

an  algebraical  function  of  u,  or  of  e  "  (where  &>  is  suitably  chosen),  or  of  g>  (u), 
where  the  periods  of  @(u)  are  suitably  chosen,  is  established;  and  it  has 
incidentally  been  established  —  it  is,  indeed,  essential  to  the  derivation  of  the 
theorem  —  that  a  function,  which  has  an  algebraical  addition-theorem,  has  only 
a  finite  number  of  values  for  a  given  argument. 

It  is  easy  to  see  that  the  first  derivative  has  only  a  finite  number  of  values 
for  a  given  argument;  for  the  elimination  of  U  between  the  algebraical 
equations 


,  , 

leads  to  an  equation  in  U'  of  the  same  finite  degree  as  G  in  U. 

Further,  it  is  now  easy  to  see  that  if  the  analytical  function  <£  (u),  which 
possesses  an  algebraical  addition-theorem,  be  uniform,  then  it  is  a  rational 

iiru 

function  either  of  u,  or  of  e  w  ,  or  of  $>  (u)  and  $'  (u)  ;  and  that  any  uniform 
function,  which  is  transcendental  in  the  sense  of  §  47  and  which  possesses  an 
algebraical  addition-theorem,  is  either  a  simply-periodic  function  or  a  doubly- 
periodic  function. 

The  following  examples  will  illustrate  some  of  the  inferences  in  regard  to  the  number 
of  values  of  <p  (u  +  v)  arising  from  series  of  values  for  u  and  v. 

Ex.  I.     Let  U=u*  +  (2u+l)*. 

Evidently  m,  the  number  of  values  of  U  for  a  value  of  u,  is  4  ;  and,  as  the  rationalised 
form  of  the  equation  is 


the  value  of  p,  being  the  number  of  values  of  u  for  a  given  value  of  U,  is  2.     Thus  the 
equation  in    W  should  be,  by  §   151,  of  degree  (4.22  —  )  16. 

This  equation  is  n  {3  (  W2  -  U2  -  F2)  +  1  -  2kr}  =  0, 

HI 
where  kr  is  any  one  of  the  eight  values  of 

W(2W*-I)*+U(2U*-l$+V(2V*-l)*; 

'     • 

an  equation,  when  rationalised,  of  the  16th  degree  in    W. 

Ex.  2.    Let  U=cosu. 

Evidently  m  =  l;  the  values  of  u  for  a  given  value  of  U  are  contained  in  the  double 
series  u  +  2irn,  -u  +  2irn,  for  all  values  of  n  from  -QO  to  +GO.     The  values  of  u  +  v  are 
,  that  is,  u  +  v  +  27rp;   -u  +  27rn+v  +  2irm,  that  is,    -u  +  v  +  2-n-p  ; 
,  that  is,  u-v  +  ^Trp;   -u  +  2irn-v  +  2irm,  that  is,    -u-v  +  Znp, 


*  The  theorem  has  been  used  by  Schwarz,  Ges.  Werke,  t.  ii,  pp.  260—268,  in  determining  all 
the  families  of  plane  isothermic  cirrves  which  are  algebraical  curves,  an  'isothermic'  curve  being 
of  the  form  u  =  c,  where  w  is  a  function  satisfying  the  potential-equation 


154.]  THE    DIFFERENTIAL    EQUATION  309 

to  that  the  number  of  series  of  values  of  u+v  is  four,  each  series  being  simply-infinite. 
It  might  thus  be  expected  that  the  equation  between    U,    V,    W  would  be   of  degree 

4  =  )  4  in    W ;  but  it  happens  that 

cos  (u  +  v)=cos(  -u-v), 
and  so  the  degree  of  the  equation  in  W  is  reduced  to  half  its  degree.     The  equation  is 

W2  -  2  WU  V+  U2  +  V2  -  1  =  0. 

Ex.  3.     Let  U=&iiu. 

Evidently  m  =  l;  and  there  are  two  doubly-infinite  series  of  values  of  u  determined 
by  a  given  value  of  U,  having  the  form  u  +  2ma>  +  2m'<o',  o>  -  w  +  2mo>  +  2m  V.  Hence  the 
values  of  u  +  v  are 

=        u+v  (mod.  2c0,  2o>') ;   =  ca-u  +  v  (mod.  2«,  2«') ; 
=  ca  +  u-v(mod.  2o>,  2<o') ;   =    -u-v  (mod.  2o>,  2&>') ; 

four  in  number.      The  equation  may  therefore  be  expected  to  be  of  the  fourth  degree 
in   W;  it  is 

4  (1  -  6T2)  (1  -  F2)  (1  -  IF2)  =  (2  -  U2-  F2-  IF2 +£2*7272  W2^ 

155.  But  it  must  not  be  supposed  that  any  algebraical  equation  between 
U,  V,  W,  which  is  symmetrical  in  U  and  V,  is  one  necessarily  implying  the 
representation  of  an  algebraical  addition-theorem.  Without  entering  into  a 
detailed  investigation  of  the  formal  characteristics  of  the  equations  that  are 
suitable,  a  latent  test  is  given  by  implication  in  the  following  theorem,  also 
due  to  Weierstrass  : — 

If  an  analytical  function  possess  an  algebraical  addition-theorem,  an 
algebraical  equation  involving  the  function  and  its  first  derivative  with  regard 
to  its  argument  exists ;  and  the  coefficients  in  this  equation  do  not  involve  the 
argument  of  the  function. 

The  proposition  might  easily  be  derived  by  assuming  the  preceding 
proposition,  and  applying  the  known  results  relating  to  the  algebraical 
dependence  between  those  functions,  the  types  of  which  are  suited  to  the 
representation  of  the  functions  in  question,  and  their  derivatives ;  we  shall, 
however,  proceed  more  directly  from  the  equation  expressing  the  algebraical 
addition-theorem  in  the  form 

G(U,V,  F)  =  0, 

which  may  be  regarded  as  a  rationally  irreducible  equation. 
Differentiating  with  regard  to  u,  we  have 

WU'+MW^Q 

dUL  +dW  ' 
and  similarly,  with  regard  to  v,  we  have 

a>+     *<=<>, 

from  which  it  follows  that 


310  EXPRESSION   OF  [155. 

This  equation*  will,  in  general,  involve  W;  in  order  to  obtain  an  equation 
free  from  W,  we  eliminate  W  between 

n       A         a  ^^  rr/       d6r  Tr/ 

G  =  0  and  ^j-  U'  =    „  V  , 

oil  ov 

the  elimination  being  possible  because  both  equations  are  of  finite  degree; 
and  thus  in  any  case  we  have  an  algebraical  equation  independent  of  W  and 
involving  U,  U',  V,  V. 

Not  more  than  one  equation  can  arise  by  assigning  various  values  to  v,  a 
quantity  that  is  independent  of  u  ;  for  we  should  have  either  inconsistent 
equations  or  simultaneous  equations  which,  being  consistent,  determine  a! 
limited  number  of  values  of  U  and  U'  for  all  values  of  u,  that  is,  only  a 
number  of  constants.  Hence  there  can  be  only  one  equation,  obtained  by 
assigning  varying  values  to  v;  and  this  single  equation  is  the  algebraical 
equation  between  the  function  and  its  first  derivative,  the  coefficients  being 
independent  of  the  argument  of  the  function. 

Note.  A  test  of  suitability  of  an  algebraical  equation  G  —  0  between 
three  variables  U,  V,  W  to  represent  an  addition-theorem  is  given  by  the 
condition  that  the  elimination  of  W  between 

G-Q  and   U'^-V  — 

dU~      dV 

leads  to  only  a  single  equation  between  U  and  U'  for  different  values  of  V 
and  V. 

Ex.     Consider  the  equation 

(Z-U-  V-  W)*-4(1-U}(1-  F)(l-  F)  =  0. 
The  deduced  equation  involving  U1  and   V  is 

(2FTF-  V-  W+  U}  U'  =  (2UW-  U-  W+  V)  V, 

,  th-it  W          (V-U}(V'+U'} 

=  (SV~lTUr 

The  elimination  of  W  is  simple.     We  have 


_ 

(27-1)  U'-(2U-\)  F" 

F    U'-l-U   V' 


utd  2     U     V     W-« 

( 

Neglecting  4  (F+  U—  1)  =  0,  which  is  an  irrelevant  equation,  arid  multiplying  by 
(2F—  1)  U'  —  (2U—l)  F',  which  is  not  zero  unless  the  numerator  also  vanish,  and  this 
would  make  both  U'  and  V  zero,  we  have 

(  F+  U-  1)  {(1  -  F)  U'  -  (1  -  U}  F'}  2  =  (1  -  U)  (1  -  F)  (  U'  -  F')  (2  F-  1)  U'  -  (2  U-  1)  F'}, 
and  therefore  V(U-V}(1-  V]  (7'2+  U(  F-  U}  (1  -  U}  F'2  =  0. 

It  is  permissible  to  adopt  any  subsidiary  irrational  or  non-algebraical  form  as  the  equivalent 
of  G  =  0,  provided  no  special  limitation  to  the  subsidiary  form  be  implicitly  adopted.  Thus,  if  W 
can  be  expressed  explicitly  in  terms  of  U  and  F,  this  resoluble  (but  irrational)  equivalent  of  the 
equation  often  leads  rapidly  to  the  equation  between  U  and  its  derivative. 


155.]  THE   ADDITION-THEOREM  311 

When  the  irrelevant  factor  U-  V  is  neglected,  this  equation  gives 

U'*  F'2 

U(l-U}~  V(l  -  V)  ' 

the  equation  required  :  and  this,  indeed,  is  the  necessary  form  in  which  the  equation 
involving  U  and  U'  arises  in  general,  the  variables  being  combined  in  associate  pairs. 
Each  side  is  evidently  a  constant,  say  4a2  ;  and  then  we  have 


Then  the  value  of  U  is  sin2  (aM+/3),  the  arbitrary  additive  constant  of  integration 
being  /3  ;  by  substitution  in  the  original  equation,  (3  is  easily  proved  to  be  zero. 

156.     Again,  if  the  elimination  between 

a  -  o  and  —  U'  -  —  V 

aduu  ~wv 

be  supposed  to  be  performed  by  the  ordinary  algebraical  process  for  finding 

o/~y  o/^r 

the  greatest  common  measure  of  G  and  U'  %Tf  —  V  %-\r>  regarded  as  functions 

of  W,  the  final  remainder  is  the  eliminant  which,  equated  to  zero,  is  the 
differential  equation  involving  U,  U',  V,  F';  and  the  greatest  common  measure, 
equated  to  zero,  gives  the  simplest  equation  in  virtue  of  which  the  equations 

G  =  0  and  ^y  U'  =  _-^  V  subsist.     It  will  be  of  the  form 
oil  ov 

f(W,U,V,  U',V')  =  0. 

If  the  function  have  only  one  value  for  each  value  of  the  argument,  so  that  it 
is  a  uniform  function,  this  last  equation  can  give  only  one  value  for  W',  for  all 
the  other  magnitudes  that  occur  in  the  equation  are  uniform  functions  of 
their  respective  arguments.  Since  it  is  linear  in  W,  the  equation  can  be 
expressed  in  the  form 

W  =  R(U,  V,  U',  V'\ 

where  R  denotes  a  rational  function.     Hence*  :  — 

A  uniform  analytical  function  (f>  (u),  which  possesses  an  algebraical 
addition-theorem,  is  such  that  (f>  (u  +  v)  can  be  expressed  rationally  in  terms 
of  $  (u),  <£'  (w),  $  (v)  and  <j>  (v). 

It  need  hardly  be  pointed  out  that  this  result  is  not  inconsistent  with  the 
fact  that  the  algebraical  equation  between  (£  (u  +  v),  (f>  (u)  and  <f>  (v)  does  not, 
in  general,  express  $(u  +  v)  as  a  rational  function  of  (f>  (u)  and  <f>(v).  And  it 
should  be  noticed  that  the  rationality  of  the  expression  of  <£  (u  +  v)  in  terms 
of  <j)  (u),  $  (v),  (/>'  (w),  $  (v)  is  characteristic  of  functions  with  an  algebraical 
addition-theorem.  Instances  do  occur  of  functions  such  that  <j)(u  +  v)  can  be 
expressed,  not  rationally,  in  terms  of  <£  (u),  </>  (v),  </>'  (u),  </>'  (v)  ;  they  do  not 
possess  an  algebraical  addition-theorem.  Such  an  instance  is  furnished  by 
%(u)',  the  expression  of  £(u  +  v),  given  in  Ex.  3  of  §  131,  can  be  modified  so  ' 
as  to  have  the  form  indicated. 

*  The  theorem  is  due  to  Weierstrass  ;  see  Schwarz,  §  2,  (I.e.  in  note  to  p.  297). 


CHAPTER   XIV. 

CONNECTION  OF  SURFACES. 

157.  IN  proceeding  to    the    discussion   of  multiform   functions,  it  was 
stated  (§  100)  that  there  are  two  methods  of  special  importance,  one  of  which 
is  the  development  of  Cauchy's  general  theory  of  functions  of  complex  vari 
ables  and  the   other  of  which  is   due  to  Riemann.     The  former  has  been 
explained    in    the    immediately  preceding   chapters ;    we    now   pass    to    the 
consideration  of  Riemann's  method.     But,  before  actually  entering  upon  it, 
there  are  some  preliminary  propositions  on  the  connection  of  surfaces  which 
must  be  established ;  as  they  do  not  find  a  place  in  treatises  on  geometry,  an 
outline   will  be  given   here   but  only   to  that  elementary  extent   which   is 
necessary  for  our  present  purpose. 

In  the  integration  of  meromorphic  functions,  it  proved  to  be  convenient 
to  exclude  the  poles  from  the  range  of  variation  of  the  variable  by  means  of 
infinitesimal  closed  simple  curves,  each  of  which  was  thereby  constituted  a 
limit  of  the  region  :  the  full  boundary  of  the  region  was  composed  of  the 
aggregate  of  these  non-intersecting  curves. 

Similarly,  in  dealing  with  some  special  cases  of  multiform  functions,  it 
proved  convenient  to  exclude  the  branch-points  by  means  of  infinitesimal 
curves  or  by  loops.  And,  in  the  case  of  the  fundamental  lemma  of  §  16,  the 
region  over  which  integration  extended  was  considered  as  one  which  possibly 
had  several  distinct  curves  as  its  complete  boundary. 

These  are  special  examples  of  a  general  class  of  regions,  at  all  points 
within  the  area  of  which  the  functions  considered  are  monogeiiic,  finite,  and 
continuous  and,  as  the  case  may  be,  uniform  or  multiform.  But,  important 
as  are  the  classes  of  functions  which  have  been  considered,  it  is  necessary  to 
consider  wider  classes  of  multiform  functions  and  to  obtain  the  regions  which 
are  appropriate  for  the  representation  of  the  variation  of  the  variable  in  each 
case.  The  most  conspicuous  examples  of  such  new  functions  are  the  algebraic 
functions,  adverted  to  in  §§  94 — 99  ;  and  it  is  chiefly  in  view  of  their  value 
and  of  the  value  of  functions  dependent  upon  them,  as  well  as  of  the  kind  of 
surface  on  which  their  variable  can  be  simply  represented,  that  we  now 
proceed  to  establish  some  of  the  topological  properties  of  surfaces  in  general. 

158.  A  surface  is  said  to  be  connected  when,  from  any  point  of  it  to  any 
other  point  of  it,  a  continuous  line  can  be  drawn  without  passing  out  of  the 


158.] 


EXAMPLES   OF   CONNECTED   SURFACES 


313 


surface.  Thus  the  surface  of  a  circle,  that  of  a  plane  ring  such  as  arises  in 
Lambert's  Theorem,  that  of  a  sphere,  that  of  an  anchor-ring,  are  connected 
surfaces.  Two  non-intersecting  spheres,  not  joined  or  bound  together  in  any 
manner,  are  not  a  connected  surface  but  are  two  different  connected  surfaces. 
It  is  often  necessary  to  consider  surfaces,  which  are  constituted  by  an 
aggregate  of  several  sheets ;  but,  in  order  that  the  surface  may  be  regarded 
as  connected,  there  must  be  junctions  between  the  sheets. 

One  of  the  simplest  connected  surfaces  is  such  a  plane  area  as  is  enclosed 
and  completely  bounded  by  the  circumference  of  a  circle.  All  lines  drawn  in 
it  from  one  internal  point  to  another  can  be  deformed  into  one  another ;  any 
simple  closed  line  lying  entirely  within  it  can  be  deformed  so  as  to  be 
evanescent,  without  in  either  case  passing  over  the  circumference ;  and  any 
simple  line  from  one  point  of  the  circumference  to  another,  when  regarded  as 
an  impassable  barrier,  divides  the  surface  into  two  portions.  Such  a  surface 
is  called*  simply  connected. 

The  kind  of  connected  surface  next  in  point  of  simplicity  is  such  a  plane 
area  as  is  enclosed  between  and  is  completely  bounded  by  the  circumferences 
of  two  concentric  circles.  All  lines  in  the  surface 
from  one  point  to  another  cannot  necessarily  be 
deformed  into  one  another,  e.g.,  the  lines  z0az  and 
zj)z;  a  simple  closed  line  cannot  necessarily  be 
deformed  so  as  to  be  evanescent  without  crossing 
the  boundary,  e.g.,  the  line  az^bza ;  and  a  simple 
line  from  a  point  in  one  part  of  the  boundary  to 
a  point  in  another  and  different  part  of  the 
boundary,  such  as  a  line  AB,  does  not  divide  the 
surface  into  two  portions  but,  set  as  an  impassable  barrier,  it  makes  the 
surface  simply  connected. 

Again,  on  the  surface  of  an  anchor-ring,  a  closed  line  can  be  drawn  in 
two  essentially  distinct  ways,  abc,  cib'c',  such 
that  neither  can  be  deformed  so  as  to  be  evanes 
cent  or  so  as  to  pass  continuously  into  the  other. 
If  abc  be  made  the  only  impassable  barrier,  a 
line  such  as  afty  cannot  be  deformed  so  as  to  be 
evanescent ;  if  ab'c'  be  made  the  only  impassable 
barrier,  the  same  holds  of  a  line  such  as  a/3'y'. 
In  order  to  make  the  surface  simply  connected, 
two  impassable  barriers,  such  as  abc  and  ab'c', 
must  be  set. 

Surfaces,  like  the  flat  ring  or  the  anchor- 


Fig.  35. 


Fig.   36. 


*  Sometimes  the  term  vionadelphic  is  used.     The  German  equivalent  is  einfach  ziisammen- 
hangend. 


314 


CROSS-CUTS   AND   LOOP-CUTS 


[158. 


ring,  are  called*  multiply  connected]  the  establishment  of  barriers  has  made  it 
possible,  in  each  case,  to  modify  the  surface  into  one  which  is  simply  connected. 

159.  It  proves  to  be  convenient  to  arrange  surfaces  in  classes  according 
to  the  character  of  their  connection ;  and  these  few  illustrations  suggest  that 
the  classification  may  be  made  to  depend,  either  upon  the  resolution  of  the 
surface,  by  the  establishment  of  barriers,  into  one  that  is  simply  connected, 
or  upon  the  number  of  what  may  be  called  independent  irreducible  circuits. 
The  former  mode — that  of  dependence  upon  the  establishment  of  barriers — 
will  be  adopted,  thus  following  Biemann-f- ;  but  whichever  of  the  two  modes 
be  adopted  (and  they  are  not  necessarily  the  only  modes)  subsequent  de 
mands  require  that  the  two  be  brought  into  relation  with  one  another. 

The  most  effective  way  of  securing  the  impassability  of  a  barrier  is  to 
suppose  the  surface  actually  cut  along  the  line  of  the  barrier.  Such  a  section 
of  a  surface  is  either  a  cross-cut  or  a  loop-cut. 

If  the  section  be  made  through  the  interior  of  the  surface  from  one  point 


Fig.  37. 

of  the  boundary  to  another  point  of  the  boundary,  without  intersecting  itself 
or  meeting  the  boundary  save  at  its  extremities,  it  is  called  a  cross-cut\. 
Every  part  of  it,  as  it  is  made,  is  to  be  regarded  as  boundary  during  the 
formation  of  the  remainder ;  and  any  cross-cut,  once  made,  is  to  be  regarded 
as  boundary  during  the  formation  of  any  cross-cut  subsequently  made. 
Illustrations  are  given  in  Fig.  37. 

The  definition  and  explanation  imply  that  the  surface  has  a  boundary. 
Some  surfaces,  such  as  a  complete  sphere  and  a  complete  anchor-ring,  do  not 
possess  a  boundary;  but,  as  will  be  seen  later  (§§  163,  168)  from  the 
discussion  of  the  evanescence  of  circuits,  it  is  desirable  to  assign  some 
boundary  in  order  to  avoid  merely  artificial  difficulties  as  to  the  numerical 

*  Sometimes  the  term  polyadc.lphic  is  used.     The  German  equivalent  is  mehrfach  zusammen- 
Mngcnd. 

t  "  Grundlagen  fur  eine  allgemeine  Theorie  der  Functionen  einer  veriindeiiichen  complexen 
Grosse,"  Eiemann's  Gesammelte  Werke,  pp.  9 — 12;   "Theorie  der  Abel'schen  Functionen,"  ib.,/ 
pp.  84—89.     When  reference  to  either  of  these  memoirs  is  made,  it  will  be  by  a  citation "et  ih^ 
page  or  pages  in  the  volume  of  lliemann's  Collected  Works. 

£  This  is  the  equivalent  used  for  the  German  word  Querschnitt ;  French  writers  use  Section, 
and  Italian  writers  use  Trasversale  or  Taglio  trasversale. 


159.]  CONNECTION   DEFINED  315 

expression  of  the  connection.  This  assignment  usually  is  made  by  taking  for 
the  boundary  of  a  surface,  which  otherwise  has  no  boundary,  an  infinitesimal 
closed  curve,  practically  a  point;  thus  in  the  figure  of  the  anchor-ring 
(Fig.  36)  the  point  a  is  taken  as  a  boundary,  and  each  of  the  two  cross-cuts 
begins  and  ends  in  a. 

If  the  section  be  made  through  the  interior  of  the  surface  from  a  point 
not  on  the  boundary  and,  without  meeting  the  boundary  or  crossing  itself, 
return  to  the  initial  point,  (so  that  it  has  the  form  of  a  simple  curve  lying 


Fig.  38. 

entirely  in  the  surface),  it  is  called*  a  loop-cut.  Thus  a  piece  can  be  cut 
out  of  a  bounded  spherical  surface  by  a  loop-cut  (Fig.  38) ;  but  it  does 
not  necessarily  give  a  separate  piece  when  made  in  the  surface  of  an 
anchor-ring. 

It  is  evident  that  both  a  cross-cut  and  a  loop-cut  furnish  a  double 
boundary-edge  to  the  whole  aggregate  of  surface,  whether  consisting  of  two 
pieces  or  of  only  one  piece  after  the  section. 

Moreover,  these  sections  represent  the  impassable  barriers  of  the  pre 
liminary  explanations ;  and  no  specified  form  was  assigned  to  those  barriers. 
It  is  thus  possible,  within  certain  limits,  to  deform  a  cross-cut  or  a  loop-cut 
continuously  into  a  closely  contiguous  and  equivalent  position.  If,  for 
instance,  two  barriers  initially  coincide  over  any  finite  length,  one  or  other 
can  be  slightly  deformed  so  that  finally  they  intersect  only  in  a  point ;  the 
same  modification  can  therefore  be  made  in  the  sections. 

The  definitions  of  simple  connection  and  of  multiple  connection  will  nowf* 
be  as  follows : — 

A  surface  is  simply  connected,  if  it  be  resolved  into  two  distinct  pieces  by 
every  cross-cut;  but  if  there  be  any  cross-cut,  which  does  not  resolve  it  into 
distinct  pieces,  the  surface  is  multiply  connected. 

160.  Some  fundamental  propositions,  relating  to  the  connection  of 
surfaces,  may  now  be  derived. 

*  This  is  the  equivalent  used  for  the  German  word  Riickkehrsclmitt ;  French  writers  use  the 
word  Retroscction. 

t  Other  definitions  will  be  required,  if  the  classification  of  surfaces  be  made  to  depend  on 
methods  other  than  resolution  by  sections. 


316  RESOLUTION   BY   CROSS-CUTS  [160. 

I.  Each  of  the  two  distinct  pieces,  into  which  a  simply  connected  surface  S 
is  resolved  by  a  cross-cut,  is  itself  simply  connected. 

If  either  of  the  pieces,  made  by  a  cross-cut  ab,  be  not  simply  connected, 
then  some  cross-cut  cd  must  be  possible  which  will  not  resolve  that  piece  into 
distinct  portions. 

If  neither  c  nor  d  lie  on  ab,  then  the  obliteration  of  the  cut  ab  will  restore 
the  original  surface  8,  which  now  is  not  resolved  by  the  cut  cd  into  distinct 
pieces. 

If  one  of  the  extremities  of  cd,  say  c,  lie  on  ab,  then  the  obliteration  of  the 
portion  cb  will  change  the  two  pieces  into  a  single  piece  which  is  the  original 
surface  8;  and  8  now  has  a  cross-cut  acd,  which  does  not  resolve  it  into 
distinct  pieces. 

If  both  the  extremities  lie  on  ab,  then  the  obliteration  of  that  part  of  ab 
which  lies  between  c  and  d  will  change  the  two  pieces  into  one ;  this  is  the 
original  surface  8,  now  with  a  cross-cut  acdb,  which  does  not  resolve  it  into 
distinct  pieces. 

These  are  all  the  possible  cases  should  either  of  the  distinct  pieces  of  8 
not  be  simply  connected ;  each  of  them  leads  to  a  contradiction  of  the  simple 
connection  of  8',  therefore  the  hypothesis  on  which  each  is  based  is  untenable, 
that  is,  the  distinct  pieces  of  8  in  all  the  cases  are  simply  connected. 

COROLLARY  1.  A  singly  connected  surface  is  resolved  by  n  cross-cuts  into 
Ti+1  distinct  pieces,  each  simply  connected;  and  an  aggregate  of  m  simply 
connected  surfaces  is  resolved  by  n  cross-cuts  into  n  -f  m  distinct  pieces  each 
simply  connected. 

COROLLARY  2.  A  surface  that  is  resolved  into  two  distinct  simply  con 
nected  pieces  by  a  cross-cut  is  simply  connected  before  the  resolution. 

COROLLARY  3.  //  a  multiply  connected  surface  be  resolved  into  two 
different  pieces  by  a  cross-cut,  both  of  these  pieces  cannot  be  simply  connected. 

We  now  come  to  a  theorem*  of  great  importance  : — 

II.  If  a  resolution  of  a  surface  by  m  cross-cuts  into  n  distinct  simply 
connected  pieces  be  possible,  and  also  a  different  resolution  of  the  same  surface  by 
fjb  cross-cuts  into  v  distinct  simply  connected  pieces,  then  m  —  n  =  fj,  —  v. 

Let  the  aggregate  of  the  n  pieces  be  denoted  by  8  and  the  aggregate  of 
the  v  pieces  by  2 :  and  consider  the  effect  on  the  original  surface  of  a  united 
system  of  in  +  p  simultaneous  cross-cuts  made  up  of  the  two  systems  of  the 
m  and  of  the  /j,  cross-cuts  respectively.  The  operation  of  this  system  can  be 
carried  out  in  two  ways :  (i)  by  effecting  the  system  of  /u,  cross-cuts  on  8  and 

*  The  following  proof  of  this  proposition  is  substantially  due  to  Neumann,  p.  157.  Another 
proof  is  given  by  Riemann,  pp.  10,  11,  and  is  amplified  by  Durege,  Elemente  der  Theorie  der 
Functional,  pp.  183 — 190  ;  and  another  by  Lippich,  see  Durege,  pp.  190 — 197. 


160.]  CONNECTIVITY  317 

(ii)  by  effecting  the  system  of  m  cross-cuts  on  2 :  with  the  same  result  on  the 
original  surface. 

After  the  explanation  of  §  159,  we  may  justifiably  assume  that  the  lines 
of  the  two  systems  of  cross-cuts  meet  only  in  points,  if  at  all :  let  8  be  the 
number  of  points  of  intersection  of  these  lines.  Whenever  the  direction  of  a 
cross-cut  meets  a  boundary  line,  the  cross-cut  terminates ;  and  if  the  direction 
continue  beyond  that  boundary  line,  that  produced  part  must  be  regarded  as 
a  new  cross-cut. 

Hence  the  new  system  of  /u,  cross-cuts  applied  to  S  is  effectively  equiva 
lent  to  (j,  +  &  new  cross-cuts.  Before  these  cuts  were  made,  S  was  composed 
of  n  simply  connected  pieces ;  hence,  after  they  are  applied,  the  new  arrange 
ment  of  the  original  surface  is  made  up  of  n  +  (/j,  +  8)  simply  connected 
pieces. 

Similarly,  the  new  system  of  m  cross-cuts  applied  to  2  will  give  an 
arrangement  of  the  original  surface  made  up  of  v  +  (m  +  8)  simply  connected 
pieces.  These  two  arrangements  are  the  same :  and  therefore 

n  +  fj,  +  8  —  v  +  in  +  8, 
so  that  m  —  n  =  p  —  v. 

It  thus  appears  that,  if  by  any  system  of  q  cross-cuts  a  multiply  connected 
surface  be  resolved  into  a  number  p  of  pieces  distinct  from  one  another  and 
all  simply  connected,  the  integer  q  —  p  is  independent  of  the  particular 
system  of  the  cross-cuts  and  of  their  configuration.  The  integer  q—p  is 
therefore  essentially  associated  with  the  character  of  the  multiple  connection 
of  the  surface  :  and  its  invariance  for  a  given  surface  enables  us  to  arrange 
surfaces  according  to  the  value  of  the  integer. 

No  classification  among  the  multiply  connected  surfaces  has  yet  been 
made :  they  have  merely  been  defined  as  surfaces  in  which  cross-cuts  can  be 
made  that  do  not  resolve  the  surface  into  distinct  pieces. 

It  is  natural  to  arrange  them  in  classes  according  to  the  number  of  cross 
cuts  which  are  necessary  to  resolve  the  surface  into  one  of  simple  connection 
or  a  number  of  pieces  each  of  simple  connection. 

For  a  simply  connected  surface,  no  such  cross-cut  is  necessary:  then 
q  =  0,  p=l,  and  in  general  q  —  p  =  —  l.  We  shall  say  that  the  connectivity* 
is  unity.  Examples  are  furnished  by  the  area  of  a  plane  circle,  and  by  a 
spherical  surface  with  one  hole^. 

A  surface  is  called  doubly- connected  when,  by  one  appropriate  cross-cut, 
the  surface  is  changed  into  a  single  surface  of  simple  connection  :  then  q  =  1, 
p  =  1  for  this  particular  resolution,  and  therefore  in  general,  q—p  =  Q.  We 

*  Sometimes  order  of  connection,  sometimes  adelphic  order ;  the  German  word,  that  is  used, 
is  Grundzahl. 

+  The  hole  is  made  to  give  the  surface  a  boundary  (§  163). 


318  EFFECT   OF   CROSS-CUTS  [160. 

shall  say  that  the  connectivity  is  2.     Examples  are  furnished  by  a  plane  ring 
and  by  a  spherical  surface  with  two  holes. 

A  surface  is  called  triply-connected  when,  by  two  appropriate  cross-cuts, 
the  surface  is  changed  into  a  single  surface  of  simple  connection :  then  q  =  2, 
p  =  l  for  this  particular  resolution  and  therefore,  in  general,  q  —  p  =  l.  We 
shall  say  that  the  connectivity  is  3.  Examples  are  furnished  by  the  surface 
of  an  anchor- ring  with  one  hole  in  it*,  and  by  the  surfaces -f-  in  Figure  39,  the 
surface  in  (2)  not  being  in  one  plane  but  one  part  beneath  another. 


Fig.  39. 

And,  in  general,  a  surface  will  be  said  to  be  ^V-ply  connected  or  its 
connectivity  will  be  denoted  by  N,  if,  by  N  —  1  appropriate  cross-cuts,  it  can 
be  changed  into  a  single  surface  that  is  simply  connected  |.  For  this 
particular  resolution  q  =  N—\,  p  =  l:  and  therefore  in  general 

q-p  =  N-2, 
or  N  =  q-p  +  2. 

Let  a  cross-cut  I  be  drawn  in  a  surface  of  connectivity  N.  There  are 
two  cases  to  be  considered,  according  as  it  does  not  or  does  divide  the  surface 
into  distinct  pieces. 

First,  let  the  surface  be  only  one  piece  after  I  is  drawn  :  and  let  its 
connectivity  then  be  N'.  If  in  the  original  surface  q  cross-cuts  (one  of 
which  can,  after  the  preceding  proposition,  be  taken  to  be  I)  be  drawn 
dividing  the  surface  into  p  simply  connected  pieces,  then 

N  =  q-p+  2. 

To  obtain  these  p  simply  connected  pieces  from  the  surface  after  the  cross-cut 
I,  it  is  evidently  sufficient  to  make  the  q  —  1  original  cross-cuts  other  than  I  ; 
that  is,  the  modified  surface  is  such  that  by  q  —  1  cross-cuts  it  is  resolved  into 
p  simply  connected  pieces,  and  therefore 


Hence  N'  =  N  —  1,  or  the  connectivity  of  the  surface  is  diminished  by  unity. 

*  The  hole  is  made  to  give  the  surface  a  boundary  (§  163). 

t  Riemann,  p.  89. 

J  A  few  writers  estimate  the  connectivity  of  such  a  surface  as  N-  1,  the  same  as  the  number 
of  cross-cuts  which  can  change  it  into  a  single  surface  of  the  simplest  rank  of  connectivity  :  the 
estimate  in  the  text  seems  preferable. 


160.] 


ON   THE   CONNECTIVITY 


319 


Secondly,  let  the  surface  be  two  pieces  after  I  is  drawn,  of  connectivities 
Ni  and  N2  respectively.  Let  the  appropriate  JVj  —  1  cross-cuts  in  the  former, 
and  the  appropriate  N2  —  1  in  the  latter,  be  drawn  so  as  to  make  each  a 
simply  connected  piece.  Then,  together,  there  are  two  simply  connected 
pieces. 

To  obtain  these  two  pieces  from  the  original  surface,  it  will  suffice  to 
make  in  it  the  cross-cut  I,  the  Ni  —  I  cross-cuts,  and  the  N2—l  cross-cuts, 
that  is,  1  +  (Ni.  —  1)  +  (N*  —  1)  or  Nj,  +  N2  —  1  cross-cuts  in  all.  Since  these, 
when  made  in  the  surface  of  connectivity  N,  give  two  pieces,  we  have 


and  therefore 

If  one  of  the  pieces  be  simply  connected,  the  connectivity  of  the  other  is  JV; 
so  that,  if  a  simply  connected  piece  of  surface  be  cut  off  a  multiply  connected 
surface,  the  connectivity  of  the  remainder  is  unchanged.  Hence  : 

III.  If  a  cross-cut  be  made  in  a  surface  of  connectivity  N  and  if  it  do 
not  divide  it  into  separate  pieces,  the  connectivity  of  the  modified  surface  is 
N—l;  but  if  it  divide  the  surface  into  two  separate  pieces  of  connectivities  N! 
and  N«,  then  Nl  +  N2  =  N+  1. 


Illustrations  are  shewn,  in  Fig.  40,  of  the  effect  of  cross-cuts  on  the  two 
surfaces  in  Fig.  39. 

IV.  In  the  same  way  it  may  be  proved  that,  if  s  cross-cuts  be  made  in  a 
surface  of  connectivity  N  and  divide  it  into  r+l  separate  pieces  (where  r^.s) 
of  connectivities  N1}  N2,  ...,  Nr+l  respectively,  then 


a  more  general  result  including  both  of  the  foregoing  cases. 

Thus  far  we  have  been  considering  only  cross-cuts  :  it  is  now  necessary 
to  consider  loop-cuts,  so  far  as  they  affect  the  connectivity  of  a  surface  in 
which  they  are  made. 


320  EFFECT   OF   LOOP-CUTS  [160. 

A  loop-cut  is  changed  into  a  cross-cut,  if  from  A  any  point  of  it  a  cross-cut 
be  made  to  any  point  C  in  a  boundary-curve  of  the 
original  surface,  for  CAbdA  (Fig.  41)  is  then  evi-  /• 

dently  a  cross-cut  of  the  original  surface  ;  and  CA  is 
a  cross-cut  of  the  surface,  which  is  the  modification 
of  the  original  surface  after  the  loop-cut  has  been 
made.  Since,  by  definition,  a  loop-cut  does  not 
meet  the  boundary,  the  cross-cut  CA  does  not 
divide  the  modified  surface  into  distinct  pieces  ; 
hence,  according  as  the  effect  of  the  loop-cut  is,  \  Fi8-  41- 

or  is  not,  that  of  making  distinct  pieces,  so  will 
the  effect  of  the  whole  cross-cut  be,  or  not  be,  that  of  making  distinct  pieces. 

161.  Let  a  loop-cut  be  drawn  in  a  surface  of  connectivity  N;  as  before 
for  a  cross-cut,  there  are  two  cases  for  consideration,  according  as  the  loop-cut 
does  or  does  not  divide  the  surface  into  distinct  pieces. 

First,  let  it  divide  the  surface  into  two  distinct  pieces,  say  of  connectivities 
N!  and  N2  respectively.  Change  the  loop-cut  into  a  cross-cut  of  the  original 
surface  by  drawing  a  cross-cut  in  either  of  the  pieces,  say  the  second,  from  a 
point  in  the  course  of  the  loop-cut  to  some  point  of  the  original  boundary. 
This  cross-cut,  as  a  section  of  that  piece,  does  not  divide  it  into  distinct 
pieces:  and  therefore  the  connectivity  is  now  N?  (=  N2  —  1).  The  effect  of 
the  whole  section,  which  is  a  single  cross-cut,  of  the  original  surface  is  to 
divide  it  into  two  pieces,  the  connectivities  of  which  are  JVa  and  N2'  :  hence, 
by  S  160,  III., 


and  therefore  N1  +  Na 

If  the  piece  cut  out  be  simply  connected,  say  JVj.  =  1,  then  the  connectivity 
of  the  remainder  is  N  +  1.  But  such  a  removal  of  a  simply  connected  piece 
by  a  loop-cut  is  the  same  as  making  a  hole  in  a  continuous  part  of  the 
surface  :  and  therefore  the  effect  of  making  a  simple  hole  in  a  continuous  part 
of  a  surface  is  to  increase  by  unity  the  connectivity  of  the  surface. 

If  the  piece  cut  out  be  doubly  connected,  say  N:  =  2,  then  the  connect 
ivity  of  the  remainder  is  N,  the  same  as  the  connectivity  of  the  original 
surface.  Such  a  portion  would  be  obtained  by  cutting  out  a  piece  with  a 
hole  in  it  which,  so  far  as  concerns  the  original  surface,  would  be  the  same  as 
merely  enlarging  the  hole  —  an  operation  that  naturally  would  not  affect 
the  connectivity. 

Secondly,  let  the  loop  -cut  not  divide  the  surface  into  two  distinct  pieces  : 
and  let  N'  be  the  connectivity  of  the  modified  surface.  In  this  modified 
surface  make  a  cross-cut  k  from  any  point  of  the  loop-cut  to  a  point  of  the 
boundary:  this  does  not  divide  it  into  distinct  pieces  and  therefore  the 
connectivity  after  this  last  modification  is  N'  -I.  But  the  surface  thus 


161.]  ON   THE   CONNECTIVITY  321 

finally  modified  is  derived  from  the  original  surface  by  the  single  cross-cut, 
constituted  by  the  combination  of  k  with  the  loop-cut  :  this  single  cross-cut 
does  not  divide  the  surface  into  distinct  pieces  and  therefore  the  connectivity 
after  the  modification  is  N  —  1.  Hence 


that  is,  JV'  =  N,  or  the  connectivity  of  a  surface  is  not  affected  by  a  loop-cut 
which  does  not  divide  the  surface  into  distinct  pieces. 

Both  of  these  results  are  included  in  the  following  theorem  :  — 

V.  If  after  any  number  of  loop-cuts  made  in  a  surface  of  connectivity 
N,  there  be  r  +  1  distinct  pieces  of  surface,  of  connectivities  JV^  JV2,  ...,  Nr+lt 
then 

N,  +  N3  +  ......  +  JVr+1  =  JV+2r. 

Let  the  number  of  loop-cuts  be  s.  Each  of  them  can  be  changed  into  a 
cross-cut  of  the  original  surface,  by  drawing  in  some  one  of  the  pieces,  as  may 
be  convenient,  a  cross-cut  from  a  point  of  the  loop-cut  to  a  point  of  a 
boundary  ;  this  new  cross-cut  does  not  divide  the  piece  in  which  it  is  drawn 
into  distinct  pieces.  If  k  such  cross-cuts  (where  k  may  be  zero)  be  drawn  in 
the  piece  of  connectivity  Nm,  the  connectivity  becomes  Nm',  where 

N  '  —  N~   —  If- 

•"  m  —  •*••  m       I"  j 
r+l  r+l  r+l 

hence  2  Nm'  =  2  Nm-2k=  X  Nm  -  s. 

m=\  m-\  m=l 

We  now  have  s  cross-cuts  dividing  the  surface  of  connectivity  JV  into  r  +  l 
distinct  pieces,  of  connectivities  JV/,  JV/,  ...,  JV/,  Nr+1'  ;  and  therefore,  by 
§  160,  IV., 


so  that  JVj  +  JV2  +  .  .  .  4-  Nr+1  =  JV  +  2r. 

This  result  could  have  been  obtained  also  by  combination  and  repetition 
of  the  two  results  obtained  for  a  single  loop-cut. 

Thus  a  spherical  surface  with  one  hole  in  it  is  simply  connected  :  when 
n  —  l  other  different  holes*  are  made  in  it,  the  edges  of  the  holes  being 
outside  one  another,  the  connectivity  of  the  surface  is  increased  by  n—  1, 
that  is,  it  becomes  n.  Hence  a  spherical  surface  with  n  holes  in  it  is  n-ply 
connected. 

162.  Occasionally,  it  is  necessary  to  consider  the  effect  of  a  slit  made  in 
the  surface. 

If  the  slit  have  neither  of  its  extremities  on  a  boundary  (and  therefore  no 
point  on  a  boundary)  it  can  be  regarded  as  the  limiting  form  of  a  loop-cut 
which  makes  a  hole  in  the  surface.  Such  a  slit  therefore  (§  161)  increases  the 
connectivity  by  unity. 

*  These  are  holes  in  the  surface,  not  holes  bored  through  the  volume  of  the  sphere  ;  one  of 
the  latter  would  give  two  holes  in  the  surface. 

F-  21 


BOUNDARIES  [162. 

If  the  slit  have  one  extremity  (but  no  other  point)  on  a  boundary,  it  can 
be  regarded  as  the  limiting  form  of  a  cross-cut,  which  returns 
on  itself  as  in  the  figure,  and  cuts  off  a  single  simply  con-         / 
nected  piece.     Such  a  slit  therefore  (§  160,  III.)  leaves  the 
connectivity  unaltered. 

If  the  slit  have  both  extremities  on  boundaries,  it  ceases      \ 
to  be  merely  a  slit :  it  is  a  cross-cut  the  effect  of  which  on  Fl8-  42- 

the  connectivity  has  been  obtained.     We  do  not  regard  such 
sections  as  slits. 

163.  In  the  preceding  investigations  relative  to  cross-cuts  and  loop-cuts, 
reference  has  continually  been  made  to  the  boundary  of  the  surface  con 
sidered. 

The  boundary  of  a  surface  consists  of  a  line  returning  to  itself,  or  of  a 
system  of  lines  each  returning  to  itself.  Each  part  of  such  a  boundary-line 
as  it  is  drawn  is  considered  a  part  of  the  boundary,  and  thus  a  boundary-line 
cannot  cut  itself  and  pass  beyond  its  earlier  position,  for  a  boundary  cannot 
be  crossed:  each  boundary-line  must  therefore  be  a  simple  curve*. 

Most  surfaces  have  boundaries :  an  exception  arises  in  the  case  of  closed 
surfaces  whatever  be  their  connectivity.  It  was  stated  (§  159)  that  a 
boundary  is  assigned  to  such  a  surface  by  drawing  an  infinitesimal  simple 
curve  in  it  or,  what  is  the  same  thing,  by  making  a  small  hole.  The 
advantage  of  this  can  be  seen  from  the  simple  example  of  a  spherical 
surface. 

When  a  small  hole  is  made  in  any  surface  the  connectivity  is  increased 
by  unity :  the  connectivity  of  the  spherical  surface  after  the  hole  is  made  is 
unity,  and  therefore  the  connectivity  of  the  complete  spherical  surface 
must  be  taken  to  be  zero. 

The  mere  fact  that  the  connectivity  is  less  than  unity,  being  that  of  the 
simplest  connected  surfaces  with  which  we  have  to  deal, 
is  not  in  itself  of  importance.     But  let  us  return  for  a 
moment  to  the  suggested  method  of  determining  the 
connectivity  by  means  of  the  evanescence  of  circuits 
without  crossing  the  boundary.     When  the  surface  is 
the  complete  spherical  surface  (Fig.  43),  there  are  two 
essentially  distinct  ways  of  making  a  circuit  C  evan 
escent,  first,  by  making  it  collapse  into  the  point  a,  Fig.  43. 
secondly  by  making  it  expand  over  the  equator  and 

then  collapse  into  the  point  b.  One  of  the  two  is  superfluous :  it  introduces 
an  element  of  doubt  as  to  the  mode  of  evanescence  unless  that  mode  be 
specified a  specification  which  in  itself  is  tantamount  to  an  assignment  of 

*  Also  a  line  not  returning  to  itself  may  be  a  boundary  ;  it  can  be  regarded  as  the  limit  of  a 
simple  curve  when  the  area  becomes  infinitesimal. 


163.]  EFFECT   OF   CROSS-CUTS   ON   BOUNDARIES  323 

boundary.  And  in  the  case  of  multiply  connected  surfaces  the  absence  of 
boundary,  as  above,  leads  to  an  artificial  reduction  of  the  connectivity  by 
unity,  arising  not  from  the  greater  simplicity  of  the  surface  but  from  the 
possibility  of  carrying  out  in  two  ways  the  operation  of  reducing  any  circuit 
to  given  circuits,  which  is  most  effective  when  only  one  way  is  permissible. 
We  shall  therefore  assume  a  boundary  assigned  to  such  closed  surfaces  as  in 
the  first  instance  are  destitute  of  boundary. 

164.  The  relations  between  the  number  of  boundaries  and  the  connect 
ivity  of  a  surface  are  given  by  the  following  propositions. 

I.  The  boimdary  of  a  simply  connected  surface  consists  of  a  single  line. 

When  a  boundary  consists  of  separate  lines,  then  a  cross-cut  can  be  made 
from  a  point  of  one  to  a  point  of  another.  By  proceeding  from 
P,  a  point  on  one  side  of  the  cross-cut,  along  the  boundary 
ac...cVwe  can  by  a  line  lying  wholly  in  the  surface  reach  a 
point  Q  on  the  other  side  of  the  cross-cut :  hence  the  parts  of 
the  surface  on  opposite  sides  of  the  cross-cut  are  connected. 
The  surface  is  therefore  not  resolved  into  distinct  pieces  by  the 
cross-cut. 

A  simply  connected  surface  is  resolved  into  distinct  pieces         Fig.  44. 
by  each  cross-cut  made  in  it :  such  a  cross-cut  as  the  foregoing 
is  therefore  not  possible,  that  is,  there  are  not  separate  lines  which  make  up 
its  boundary.     It  has  a  boundary :  the  boundary  therefore  consists  of  a  single 
line. 

II.  A  cross-cut  either  increases  by  unity  or  diminishes  by  unity  the  number 
of  distinct  boundary -lines  of  a  multiply  connected  surface. 

A  cross-cut  is  made  in  one  of  three  ways :  either  from  a  point  a  of  one 
boundary-line  A  to  a,  point  b  of  another  boundary-line  B ;  or  from  a  point  a 
of  a  boundary-line  to  another  point  a'  of  the  same  boundary-line  ;  or  from  a 
point  of  a  boundary-line  to  a  point  in  the  cut  itself. 

If  made  in  the  first  way,  a  combination  of  one  edge  of  the  cut,  the 
remainder  of  the  original  boundary  A,  the  other  edge  of  the  cut  and  the 
remainder  of  the  original  boundary  B  taken  in  succession,  form  a  single 
piece  of  boundary ;  this  replaces  the  two  boundary-lines  A  and  B  which 
existed  distinct  from  one  another  before  the  cross-cut  was  made.  Hence  the 
number  of  lines  is  diminished  by  unity.  An  example  is  furnished  by  a  plane 
ring  (ii.,  Fig.  37,  p.  314). 

If  made  in  the  second  way,  the  combination  of  one  edge  of  the  cut  with 
the  piece  of  the  boundary  on  one  side  of  it  makes  one  boundary-line,  and  the 
combination  of  the  other  edge  of  the  cut  with  the  other  piece  of  the  boundary 
makes  another  boundary-line.  Two  boundary-lines,  after  the  cut  is  made, 

21—2 


324  NUMBER   OF   BOUNDARY-LINES  [164. 

replace  a  single  boundary-line,  which  existed  before  it  was  made :  hence  the 
number  of  lines  is  increased  by  unity.  Examples  are  furnished  by  the  cut 
surfaces  in  Fig.  40,  p.  319. 

If  made  in  the  third  way,  the  cross-cut  may  be  considered  as  constituted 
by  a  loop-cut  and  a  cut  joining  the  loop-cut  to  the  boundary.  The  boundary- 
lines  may  now  be  considered  as  constituted  (Fig.  41,  p.  320)  by  the  closed 
curve  ABD  and  the  closed  boundary  abda'c'e'...eca;  that  is,  there  are  now 
two  boundary-lines  instead  of  the  single  boundary-line  ce...e'c'c  in  the  uncut 
surface.  Hence  the  number  of  distinct  boundary-lines  is  increased  by  unity. 

COROLLARY.  A  loop-cut  increases  the  number  of  distinct  boundary-lines 
by  two. 

This  result  follows  at  once  from  the  last  discussion. 

III.  The  number  of  distinct  boundary-lines  of  a  surface  of  connectivity  N 
is  N  —  2k,  where  k  is  a  positive  integer  that  may  be  zero. 

Let  m  be  the  number  of  distinct  boundary-lines ;  and  let  N  —  1  appro 
priate  cross-cuts  be  drawn,  changing  the  surface  into  a  simply  connected 
surface.  Each  of  these  cross-cuts  increases  by  unity  or  diminishes  by  unity 
the  number  of  boundary-lines ;  let  these  units  of  increase  or  of  decrease  be 
denoted  by  e^  e2,  ...,  €#_!.  Each  of  the  quantities  e  is  +  1 ;  let  k  of  them  be 
positive,  and  N  —  1  —  k  negative.  The  total  number  of  boundary-lines  is 

therefore 

m  +  k-(N-l-k). 

The  surface  now  is  a  single  simply  connected  surface,  and  there  is  therefore 
only  one  boundary-line  ;  hence 

m  +  k-(N-l-k)  =  l, 
so  that  m  =  N  —  2k ; 

and  evidently  k  is  an  integer  that  may  be  zero. 

COROLLARY  1.  A  closed  surface  with  a  single  boundary-line*  is  of  odd 
connectivity. 

For  example,  the  surface  of  an  anchor-ring,  when  bounded,  is  of  con 
nectivity  3;  the  surface,  obtained  by  boring  two  holes  through  the  volume 
of  a  solid  sphere,  is,  when  bounded,  of  connectivity  5. 

If  the  connectivity  of  a  closed  surface  with  a  single  boundary  be  2p  +  1, 
the  surface  is  often  said-f-  to  be  of  class  p  (§  178,  p.  349.) 

COROLLARY  2.  If  the  number  of  distinct  boundary  lines  of  a  surface  of 
connectivity  N  be  N,  any  loop-cut  divides  the  surface  into  two  distinct  pieces. 

After  the  loop-cut  is  made,  the  number  of  distinct  boundary-lines  is 
N+2;  the  connectivity  of  the  whole  of  the  cut  surface  is  therefore  not  less 

*  See  §  159. 

t  The  German  word  is  Geschlecht ;  French  writers  use  the  word  genre,  and  Italians  genere. 


164.]  LHUILIER'S  THEOREM  325 

than  N+2.  It  has  been  proved  that  a  loop-cut,  which  does  not  divide  the 
surface  into  distinct  pieces,  does  not  affect  the  connectivity ;  hence  as  the 
connectivity  has  been  increased,  the  loop-cut  must  divide  the  surface  into 
two  distinct  pieces.  It  is  easy,  by  the  result  of  §  161,  to  see  that,  after  the 
loop-cut  is  made,  the  sum  of  connectivities  of  the  two  pieces  is  N+2,  so 
that  the  connectivity  of  the  whole  of  the  cut  surface  is  equal  to  N  +  2. 

Note.  Throughout  these  propositions,  a  tacit  assumption  has  been  made, 
which  is  important  for  this  particular  proposition  when  the  surface  is  the 
means  of  representing  the  variable.  The  assumption  is  that  the  surface  is 
bifacial  and  not  unifacial ;  it  has  existed  implicitly  throughout  all  the 
geometrical  representations  of  variability :  it  found  explicit  expression  in 
§  4  when  the  plane  was  brought  into  relation  with  the  sphere :  and  a  cut 
in  a  surface  has  been  counted  a  single  cut,  occurring  in  one  face,  though  it 
would  have  to  be  counted  as  two  cuts,  one  on  each  side,  were  the  surface 
unifacial. 

The  propositions  are  not  necessarily  valid,  when  applied  to  unifacial 
surfaces.  Consider  a  surface  made  out  of  a  long  rectangular  slip  of  paper, 
which  is  twisted  once  (or  any  odd  number  of  times)  and  then  has  its  ends 
fastened  together.  This  surface  is  of  double  connectivity,  because  one 
section  can  be  made  across  it  which  does  not  divide  it  into  separate  pieces ; 
it  has  only  a  single  boundary-line,  so  that  Prop.  III.  just  proved  does  not 
!  apply.  The  surface  is  unifacial ;  and  it  is  possible,  without  meeting  the 
boundary,  to  pass  continuously  in  the  surface  from  a  point  P  to  another 
point  Q  which  could  be  reached  merely  by  passing  through  the  material 
at  P. 

We  therefore  do  not  retain  unifacial  surfaces  for  consideration. 

165.  The  following  proposition,  substantially  due  to  Lhuilier*,  may  be 
taken  in  illustration  of  the  general  theory. 

If  a  closed  surface  of  connectivity  2N  + 1  (or  of  class  N)  be  divided  by 
circuits  into  any  number  of  simply  connected  portions,  each  in  the  form  of  a 
curvilinear  polygon,  and  if  F  be  the  number  of  polygons,  E  be  the  number  of 
edges  and  S  the  number  of  angular  points,  then 

2N=2  +  JE-F-S. 

Let  the  edges  E  be  arranged  in  systems,  a  system  being  such  that  any 
lino  in  it  can  be  reached  by  passage  along  some  other  line  or  lines  of  the 
system  ;  let  k  be  the  number  of  such  systems -f.  To  resolve  the  surface  into  a 
number  of  simply  connected  pieces  composed  of  the  F  polygons,  the  cross-cuts 
will  be  made  along  the  edges ;  and  therefore,  unless  a  boundary  be  assigned 

*  Gergonne,  Ann.  de  Math.,  t.  iii,  (1813),  pp.  181—186;  see  also  Mobius,  Ges.  Werke,  t.  ii, 
p.  468.     A  circuit  is  defined  in  §  166. 

t  The  value  of  k  is  1  for  the  proposition  and  is  greater  than  1  for  the  Corollary. 


326  LHUILIER'S  THEOREM  [165. 

to  the  surface  in  each  system  of  lines,  the  first  cut  for  any  system  will  be  a 
loop-cut.  We  therefore  take  k  points,  one  in  each  system  as  a  boundary  ; 
the  first  will  be  taken  as  the  natural  boundary  of  the  surface,  and  the 
remaining  k—\,  being  the  limiting  forms  of  k  —  1  infinitesimal  loop-cuts, 
increase  the  connectivity  of  the  surface  by  k  —  1,  that  is,  the  connectivity  now 
is  2N+k. 

The  result  of  the  cross-cuts  is  to  leave  F  simply  connected  pieces  :  hence 
Q,  the  number  of  cross-cuts,  is  given  by 


At  every  angular  point  on  the  uncut  surface,  three  or  more  polygons  are 
contiguous.  Let  Sm  be  the  number  of  angular  points,  where  m  polygons  are 
contiguous;  then 


Again,  the  number.  of  edges  meeting  at  each  of  the  S3  points  is  three,  atl 
each  of  the  $4  points  is  four,  at  each  of  the  $5  points  is  five,  and  so  on  ;  hence, 
in  taking  the  sum  3$3  +  4$4  +  5$5  +  .  .  .,  each  edge  has  been  counted  twice,  once 
for  each  extremity.     Therefore 


Consider  the  composition  of  the  extremities  of  the  cross-cuts  ;  the  number 
of  the  extremities  is  2Q,  twice  the  number  of  cross-cuts. 

Each  of  the  k  points  furnishes  two  extremities;  for  each  such  point 
is  a  boundary  on  which  the  initial  cross-cut  for  each  of  the  systems  must 
begin  and  must  end.  These  points  therefore  furnish  2k  extremities. 

The  remaining  extremities  occur  in  connection  with  the  angular  points. 
In  making  a  cut,  the  direction  passes  from  a  boundary  along  an  edge,  past 
the  point  along  another  edge  and  so  on,  until  a  boundary  is  reached  ;  so  that 
on  the  first  occasion  when  a  cross-cut  passes  through  a  point,  it  is  made  along 
two  of  the  edges  meeting  at  the  point.  Every  other  cross-cut  passing  through 
that  point  must  begin  or  end  there,  so  that  each  of  the  S3  points  will  furnish 
one  extremity  (corresponding  to  the  remaining  one  cross-cut  through  the 
point),  each  of  the  $4  points  will  furnish  two  extremities  (corresponding  to 
the  remaining  two  cross-cuts  through  the  point),  and  so  on.  The  total 
number  of  extremities  thus  provided  is 

S3  +  2St+3S5  +  ... 
Hence  2Q  =  2k  +  83  +  2St  +  3S6+  ... 


or  Q  =  k  +  E-S, 

which  combined  with         Q  =  2N  +  k  +  F  -  2, 
leads  to  the  relation        2N=2  +  E-F-S. 


165.]  CIRCUITS   ON   CONNECTED   SURFACES  327 

The  simplest  case  is  that  of  a  sphere,  when  Euler's  relation  F  +  S  =•  E  +  2 
is  obtained.  The  case  next  in  simplicity  is  that  of  an  anchor-ring,  for  which 
the  relation  is  F+  S  =  E. 

COROLLARY.  If  the  result  of  making  the  cross-cuts  along  the  various  edges 
be  to  give  the  F  polygons,  not  simply  connected  areas  but  areas  of  connectivities 
jYj  +  1,  jV2  +  l,  ...,  Np+1  respectively,  then  the  connectivity  of  the  original 
surface  is  given  by 


166.  The  method  of  determining  the  connectivity  of  a  surface  by  means 
of  a  system  of  cross-cuts,  which  resolve  it  into  one  or  more  simply  connected 
pieces,  will  now  be  brought  into  relation  with  the  other  method,  suggested 
in  §  159,  of  determining  the  connectivity  by  means  of  irreducible  circuits. 

A  closed  line  drawn  on  the  surface  is  called  a  circuit. 

A  circuit,  which  can  be  reduced  to  a  point  by'  continuous  deformation 
without  crossing  the  boundary,  is  called  reducible  ;  a  circuit,  which  cannot  be 
so  reduced,  is  called  irreducible. 

An  irreducible  circuit  is  either  (i)  simple,  when  it  cannot  without  crossing 
the  boundary  be  deformed  continuously  into  repetitions  of  one  or  more 
circuits  ;  or  (ii)  multiple,  when  it  can  without  crossing  the  boundary  be 
deformed  continuously  into  repetitions  of  a  single  circuit  ;  or  (iii)  compound, 
when  it  can  without  crossing  the  boundary  be  deformed  continuously  into 
combinations  of  different  circuits,  that  may  be  simple  or  multiple.  The 
distinction  between  simple  circuits  and  compound  circuits,  that  involve  no 
multiple  circuits  in  their  combination,  depends  upon  conventions  adopted  for 
each  particular  case. 

A  circuit  is  said  to  be  reconcileable  with  the  system  of  circuits  into  a 
combination  of  which  it  can  be  continuously  deformed. 

If  a  system  of  circuits  be  reconcileable  with  a  reducible  circuit,  the 
system  is  said  to  be  reducible. 

As  there  are  two  directions,  one  positive  and  the  other  negative,  in  which 
a  circuit  can  be  described,  and  as  there  are  possibilities  of  repetitions  and  of 
compositions  of  circuits,  it  is  clear  that  circuits  can  be  represented  by  linear 
algebraical  expressions  involving  real  quantities  and  having  merely  numerical 
coefficients. 

Thus  a  reducible  circuit  can  be  denoted  by  0. 

If  a  simple  irreducible  circuit,  positively  described,  be  denoted  by  a,  the 
same  circuit,  negatively  described,  can  be  denoted  by  —  a. 

The  multiple  circuit,  which  is  composed  of  m  positive  repetitions  of  the 
simple  irreducible  circuit  a,  would  be  denoted  by  ma  ;  but  if  the  m  repetitions 
were  negative,  the  multiple  circuit  would  be  denoted  by  —  ma. 


328  CIRCUITS  [106. 

A  compound  circuit,  reconcileable  with  a  system  of  simple  irreducible 
circuits  a1}  a2,  ...,  an  would  be  denoted  by  m1a1  +  m2a2-\-  ...  +  mnan,  where 
mj,  m2,  ...,  mn  are  positive  or  negative  integers,  being  the  net  number  of 
positive  or  negative  descriptions  of  the  respective  simple  irreducible  circuits. 

The  condition  of  the  reducibility  of  a  system  of  circuits  al,  «2,  ...,  an, 
each  one  of  which  is  simple  and  irreducible,  is  that  integers  m1}  m.2,  ...,  mn 
should  exist  such  that 

m^j  +  m2a2  +  . . .  +  mnan  =  0, 

the  sign  of  equality  in  this  equation,  as  in  other  equations,  implying  that 
continuous  deformation  without  crossing  the  boundary  can  change  into  one 
another  the  circuits,  denoted  by  the  symbols  on  either  side  of  the  sign. 

The  representation  of  any  compound  circuit  in  terms  of  a  system  of 
independent  irreducible  circuits  is  unique :  if  there  were  two  different 
expressions,  they  could  be  equated  in  the  foregoing  sense  and  this  would 
imply  the  existence  of  a  'relation 

P&  +  p.2a2  +  . . .  +pnan  =  0, 
which  is  excluded  by  the  fact  that  the  system  is  irreducible. 

Further,  equations  can  be  combined  linearly,  provided  that  the  coefficients 
of  the  combinations  be  merely  numerical. 

167.  In  order,  then,  to  be  in  a  position  to  estimate  circuits  on  a  multiply 
connected  surface,  it  is  necessary  that  an  irreducible  system  of  irreducible 
simple  circuits  should  be  known,  such  a  system  being  considered  complete 
when  every  other  circuit  on  the  surface  is  reconcileable  with  the  system. 

Such  a  system  is  not  necessarily  unique ;  and  it  must  be  proved  that,  if 
more  than  one  complete  system  be  obtainable,  any  circuit  can  be  reconciled  with 
each  system. 

First,  the  number  of  simple  irreducible  circuits  in  any  complete  system 
must  be  tlie  same  for  the  same  surface. 

Let  a1}  ...,  ap;  and  b1}  ...,  bn;  be  two  complete  systems.  Because  a1}  ..., 
ap  constitute  a  complete  system,  every  circuit  of  the  system  of  circuits  b  is 
reconcileable  with  it ;  that  is,  integers  ra#  exist,  such  that 

br  =  mlral  +  m.2ra.2  +  . . .  +  mprap, 

for  r  =  1,  2,  ...,  n.  If  n  were  >p,  then  by  combining  linearly  each  equation 
after  the  first  p  equations  with  those  p  equations,  and  eliminating  al,  ...,  ap 
from  the  set  of  p  +  1  equations,  we  could  derive  n  —p  relations  of  the  form 

M^  +  M,b2  +  . . .  +  Mnbn  =  0, 

where  the  coefficients  M,  being  determinants  the  constituents  of  which  are 
integers,  would  be  integers.  The  system  of  circuits  b  is  irreducible,  and  there 
are  therefore  no  such  relations ;  hence  n  is  not  greater  than  p. 


167.]  ON  CONNECTED  SURFACES  329 

Similarly,  by  considering  the  reconciliation  of  each  circuit  a  with  the 
irreducible  system  of  circuits  b,  it  follows  that  p  is  not  greater  than  n. 

Hence  p  and  n  are  equal  to  one  another.     And,  because  each  system  is  a 
complete  system,  there  are  integers  A  and  B  such  that 
ar  =  Arlbi  +  Ar2b.2 4-  •  •  •  +  Arnbn    (r  =  I,  ..., 
bs  =  Bg^  +  Bs.2a2  +  . . .  -I- BmOn    (s  =  l,  ..., 

The  determinant  of  the  integers  A  is  equal  to  +  1 ;  likewise  the  deter 
minant  of  the  integers  B. 

Secondly,  let  x  be  a  circuit  reconcileable  with  the  system  of  circuits  a :  it  is 
reconcileable  with  any  other  complete  system  of  circuits. 

Since  x  is  reconcileable  with  the  system  a,  integers  m1}  ...,  mn  can  be 

found  such  that 

x  =  ??i1«1  +  . . .  +  mnan. 

Any  other  complete  system  of  n  circuits  b  is  such  that  the  circuits  a  can 
be  expressed  in  the  form 

ar  =  Anbj.  +  ...  +  Arnbn ,     (r  =  1,  . . .,  n), 
where  the  coefficients  A  are  integers ;  and  therefore 

n  n  n 

x  =  b1'2  mrArl  4-  62  S  mrArz  +  . . .  +  bn  X  mrArn 

r=l  r=l  r=l 

=  gri&i  +  gr2&a  +  ~'+qnl>n, 

where  the  coefficients  q  are  integers,  that  is,  x  is  reconcileable  with  the 
complete  system  of  circuits  b. 

168.  It  thus  appears  that  for  the  construction  of  any  circuit  on  a  surface, 
it  is  sufficient  to  know  some  one  complete  system  of  simple  irreducible 
circuits.  A  complete  system  is  supposed  to  contain  the  smallest  possible 
number  of  simple  circuits :  any  one  which  is  reconcileable  with  the  rest  is 
omitted,  so  that  the  circuits  of  a  system  may  be  considered  as  independent. 
Such  a  system  is  indicated  by  the  following  theorems : — 

I.  No  irreducible  simple  circuit  can  be  drawn  on  a  simply  connected 
surface*. 

If  possible,  let  an  irreducible  circuit  G  be  drawn  in  a  simply  connected 
surface  with  a  boundary  B.  Make  a  loop-cut  along  C,  and  change  it  into  a 
cross-cut  by  making  a  cross-cut  A  from  some  point  of  C  to  a  point  of  B ; 
this  cross-cut  divides  the  surface  into  two  simply  connected  pieces,  one  of 
which  is  bounded  by  B,  the  two  edges  of  A,  and  one  edge  of  the  cut  along  C, 
and  the  other  of  which  is  bounded  entirely  by  the  cut  along  C. 

The   latter  surface   is  smaller  than  the   original  surface ;    it   is  simply 

connected  and  has  a  single  boundary.     If  an  irreducible  simple  circuit  can 

be  drawn  on  it,  we  proceed  as  before,  and  again  obtain  a  still  smaller  simply 

connected   surface.      In   this    way,    we    ultimately    obtain    an    infinitesimal 

*  All  surfaces  considered  are  supposed  to  be  bounded. 


330  RELATIONS   BETWEEN    CONNECTIVITY  [168. 

element ;  for  every  cut  divides  the  surface,  in  which  it  is  made,  into 
distinct  pieces.  Irreducible  circuits  cannot  be  drawn  in  this  element ;  and 
therefore  its  boundary  is  reducible.  This  boundary  is  a  circuit  in  a  larger 
portion  of  the  surface :  the  circuit  is  reducible  so  that,  in  that  larger  portion 
no  irreducible  circuit  is  possible  and  therefore  its  boundary  is  reducible. 
This  boundary  is  a  circuit  in  a  still  larger  portion,  and  the  circuit  is 
reducible :  so  that  in  this  still  larger  portion  no  irreducible  circuit  is  possible 
and  once  more  the  boundary  is  reducible. 

Proceeding  in  this  way,  we  find  that  no  irreducible  simple  circuit  is 
possible  in  the  original  surface. 

COROLLARY.  No  irreducible  circuit  can  be  drawn  on  a  simply  connected 
surface. 

II.  A  complete  system  of  irreducible  simple  circuits  for  a  surface  of 
connectivity  N  contains  N—  I  simple  circuits,  so  that  every  other  circuit  on  the 
surface  is  reconcileable  with  that  system. 

Let  the  surface  be  resolved  by  cross-cuts  into  a  single  simply  connected 
surface:   N—  1  cross-cuts  will  be  necessary.     Let  CD  be 
any  one  of  them :   and  let  a  and  b  be  two  points  on  the  /e 

opposite  edges  of  the  cross-cut.     Then  since  the  surface  is  L        n 

simply  connected,  a  line  can  be  drawn  in  the  surface  from 
a  to  b  without  passing  out  of  the  surface  or  without 
meeting  a  part  of  the  boundary,  that  is,  without  meeting 
any  other  cross-cut.  The  cross-cut  CD  ends  either  in  Fis- 45- 

another  cross-cut  or  in  a  boundary;  the  line  ae...fb 
surrounds  that  other  cross-cut  or  that  boundary  as  the  case  may  be :  hence, 
if  the  cut  CD  be  obliterated,  the  line  ae...fba  is  irreducible  on  the  surface  in 
which  the  other  N  —  2  cross-cuts  are  made.  But  it  meets  none  of  those  cross 
cuts;  hence,  when  they  are  all  obliterated  so  as  to  restore  the  unresolved 
surface  of  connectivity  N,  it  is  an  irreducible  circuit.  It  is  evidently  riot 
a  repeated  circuit;  hence  it  is  an  irreducible  simple  circuit.  Hence  the 
line  of  an  irreducible  simple  circuit  on  an  unresolved  surface  is  given  by 
a  line  passing  from  a  point  on  one  edge  of  a  cross-cut  in  the  resolved 
surface  to  a  point  on  the  opposite  edge. 

Since  there  are  N  -I  cross-cuts,  it  follows  that  N —1  irreducible  simple 
circuits  can  thus  be  obtained:  one  being  derived  in  the  foregoing  manner 
from  each  of  the  cross-cuts,  which  are  necessary  to  render  the  surface  simply 
connected.  It  is  easy  to  see  that  each  of  the  irreducible  circuits  on  an 
unresolved  surface  is,  by  the  cross-cuts,  rendered  impossible  as  a  circuit  on 
the  resolved  surface. 

But  every  other  irreducible  circuit  C  is  reconcileable  with  the  N—l 
circuits,  thus  obtained.  If  there  be  one  not  reconcileable  with  these  N-l 
circuits,  then,  when  all  the  cross-cuts  are  made,  the  circuit  C  is  not  rendered 


168.] 


AND   IRREDUCIBLE   CIRCUITS 


331 


impossible,  if  it  be  not  reconcileable  with  those  which  are  rendered  impossible 
by  the  cross-cuts :  that  is,  there  is  on  the  resolved  surface  an  irreducible 
circuit.  But  the  resolved  surface  is  simply  connected,  and  therefore  no 
irreducible  circuit  can  be  drawn  on  it :  hence  the  hypothesis  as  to  C,  which 
leads  to  this  result,  is  not  tenable. 

Thus  every  other  circuit  is  reconcileable  with  the  system  of  N  —  1  circuits  : 
and  therefore  the  system  is  complete*. 

This  method  of  derivation  of  the  circuits  at  once  indicates  how  far  a 
system  is  arbitrary.  Each  system  of  cross-cuts  leads  to  a  complete  system  of 
irreducible  simple  circuits,  and  vice  versa ;  as  the  one  system  is  not  unique, 
so  the  other  system  is  not  unique. 

For  the  general  question,  Jordan's  memoir,  Des  contours  traces  sur  les  surfaces, 
Liouville,  2me  Ser.,  t.  xi.,  (1866),  pp.  110—130,  may  be  consulted. 

Ex.  1.  On  a  doubly  connected  surface,  one  irreducible  simple  circuit  can  be  drawn. 
It  is  easily  obtained  by  first  resolving  the  surface  into  one  that  is  simply  connected — 
a  single  cross-cut  CD  is  effective  for  this  purpose — and  then  by  drawing  a  curve  aeb  in  the 


Fig.  46,  (i). 

surface  from  one  edge  of  the  cross-cut  to  the  other.     All  other  irreducible  circuits  on  the 
unresolved  surface  are  reconcileable  with  the  circuit  aeba. 

Ex.  2.      On  a  triply- connected  surface,  two  independent  irreducible  circuits  can  be 


Fig.  46,  (ii). 

*  If  the  number  of  independent  irreducible  simple  circuits  be  adopted  as  a  basis  for  the 
definition  of  the  connectivity  of  a  surface,  the  result  of  the  proposition  would  be  taken  as  the 
definition  :  and  the  resolution  of  the  surface  into  one,  which  is  simply  connected,  would  then  be 
obtained  by  developing  the  preceding  theory  in  the  reverse  order. 


332 


DEFORMATION 


[168. 


drawn.  Thus  in  the  figure  Cl  and  C2  will  form  a  complete  system.  The  circuits  C3  and  (74 
are  also  irreducible  :  they  can  evidently  be  deformed  into  C^  and  <72  and  reducible  circuits 
by  continuous  deformation  :  in  the  algebraical  notation  adopted,  we  have 

C3=C1  +  C2,     Ci=Cl-C.2. 

Ex.  3.     Another  example  of  a  triply  connected  surface  is  given  in  Fig.  47.     Two  irredu 
cible   simple   circuits   are    Cv   and  C%.      Another  irreducible  circuit  is  C3;   this  can  be 


Fig.  47. 

reconciled  with  Cl  and  C.2  by  drawing  the  point  a  into  coincidence  with  the  intersection 
of  Cj  and  (72,  and  the  point  c  into  coincidence  with  the  same  point. 

Ex.  4.     As  a  last  example,  consider  the  surface  of  a  solid  sphere  with  n  holes  bored 
through  it.     The  connectivity  is  2n  + 1  :  hence  2n  independent  irreducible  simple  circuits 


Fig.  48. 

can  be  drawn  on  the  surface.  The  simplest  complete  system  is  obtained  by  taking  2n 
curves :  made  up  of  a  set  of  n,  each  round  one  hole,  and  another  set  of  n,  each  through 
one  hole. 

A  resolution  of  this  surface  is  given  by  taking  cross-cuts,  one  round  each  hole  (making 
the  circuits  through  the  holes  no  longer  possible)  and  one  through  each  hole  (making  the 
circuits  round  the  holes  no  longer  possible). 

The  simplest  case  is  that  for  which  n=  1  :  the  surface  is  equivalent  to  the  anchor-ring. 

169.  Surfaces  are  at  present  being  considered  in  view  of  their  use  as  a 
means  of  representing  the  value  of  a  complex  variable.  The  foregoing  inves 
tigations  imply  that  surfaces  can  be  classed  according  to  their  connectivity ; 
and  thus,  having  regard  to  their  designed  use,  the  question  arises  as  to 
whether  all  surfaces  of  the  same  connectivity  arc  equivalent  to  one  another, 
so  as  to  be  transformable  into  one  another. 


169.]  OF  CONNECTED  SURFACES  333 

Moreover,  a  surface  can  be  physically  deformed  and  still  remain  suitable  for 
representation  of  the  variable,  provided  certain  conditions  are  satisfied.  We 
thus  consider  geometrical  transformation  as  well  as  physical  deformation  ;  but 
we  are  dealing  only  with  the  general  results  and  not  with  the  mathematical 
relations  of  stretching  and  bending,  which  are  discussed  in  treatises  on 
Analytical  Geometry*. 

It  is  evident  that  continuity  is  necessary  for  both :  discontinuity  would 
imply  discontinuity  in  the  representation  of  the  variable.  Points  that  are 
contiguous  (that  is,  separated  only  by  small  distances  measured  in  the  surface) 
must  remain  contiguous -f*:  and  one  point  in  the  unchanged  surface  must 
correspond  to  only  one  point  in  the  changed  surface.  Hence  in  the  continuous 
deformation  of  a  surface  there  may  be  stretching  and  there  may  be  bending ; 
but  there  must  be  no  tearing  and  there  must  be  no  joining. 

For  instance,  a  single  untwisted  ribbon,  if  cut,  comes  to  be  simply  connected.  If  a  twist 
through  180°  be  then  given  to  one  end  and  that  end  be  then  joined  to  the  other,  we  shall 
have  a  once- twisted  ribbon,  which  is  a  surface  with  only  one  face  and  only  one  edge; 
it  cannot  be  looked  upon  as  an  equivalent  of  the  former  surface. 

A  spherical  surface  with  a  single  hole  can  have  the  hole  stretched  and  the  surface 
flattened,  so  as  to  be  the  same  as  a  bounded  portion  of  a  plane  :  the  two  surfaces  are 
equivalent  to  one  another.  Again,  in  the  spherical  surface,  let  a  large  indentation  be 
made  :  let  both  the  outer  and  the  inner  surfaces  be  made  spherical ;  and  let  the  mouth  of 
the  indentation  be  contracted  into  the  form  of  a  long,  narrow  hole  along  a  part  of  a  great 
circle.  When  each  point  of  the  inner  surface  is  geometrically  moved  so  that  it  occupies  the 
position  of  its  reflexion  in  the  diametral  plane  of  the  hole,  the  final  form§  of  the  whole 
surface  is  that  of  a  two-sheeted  surface  with  a  junction  along  a  line  :  it  is  a  spherical 
winding-surface,  and  is  equivalent  to  the  simply  connected  spherical  surface. 

170.  It  is  sufficient,  for  the  purpose  of  representation,  that  the  two 
surfaces  should  have  a  point-to-point  transformation  :  it  is  not  necessary 
that  physical  deformation,  without  tears  or  joins,  should  be  actually  possible. 
Thus  a  ribbon  with  an  even  number  of  twists  would  be  as  effective  as  a 
limited  portion  of  a  cylinder,  or  (what  is  the  same  thing)  an  untwisted  ribbon : 
but  it  is  not  possible  to  deform  the  one  into  the  other  physically  |. 

It  is  easy  to  see  that  either  deformation  or  transformation  of  the  kind 
considered  will  change  a  bifacial  surface  into  a  bifacial  surface ;  that  it  will 
not  alter  the  connectivity,  for  it  will  not  change  irreducible  circuits  into 

*  See,  for  instance,  Frost's  Solid  Geometry,  (3rd  ed.),  pp.  342 — 352. 

t  Distances  between  points  must  be  measured  along  the  surface,  not  through  space ;  the 
distance  between  two  points  is  a  length  which  one  point  would  traverse  before  reaching  the 
position  of  the  other,  the  motion  of  the  point  being  restricted  to  take  place  in  the  surface. 
Examples  will  arise  later,  in  Biemann's  surfaces,  in  which  points  that  are  contiguous  in  space 
are  separated  by  finite  distances  on  the  surface. 

§  Clifford,  Coll.  Hath.  Papers,  p.  250. 

J  The  difference  between  the  two  cases  is  that,  in  physical  deformation,  the  surfaces  are  the 
surfaces  of  continuous  matter  and  are  impenetrable ;  while,  in  geometrical  transformation,  the 
surfaces  may  be  regarded  as  penetrable  without  interference  with  the  continuity. 


334  DEFORMATION   OF   SURFACES  [170. 

reducible  circuits,  and  the  number  of  independent  irreducible  circuits 
determines  the  connectivity:  and  that  it  will  not  alter  the  number  of  boundary 
curves,  for  a  boundary  will  be  changed  into  a  boundary.  These  are  necessary 
relations  between  the  two  forms  of  the  surface  :  it  is  not  difficult  to  see  that 
they  are  sufficient  for  correspondence.  For  if,  on  each  of  two  bifacial  surfaces 
with  the  same  number  of  boundaries  and  of  the  same  connectivity,  a  complete 
system  of  simple  irreducible  circuits  be  drawn,  then,  when  the  members  of  the 
systems  are  made  to  correspond  in  pairs,  the  full  transformation  can  be  effected 
by  continuous  deformation  of  those  corresponding  irreducible  circuits.  It 
therefore  follows  that : — 

The  necessary  and  sufficient  conditions,  that  two  bifacial  surfaces  may  be 
equivalent  to  one  another  for  the  representation  of  a  variable,  are  that  tlie  two 
surfaces  should  be  of  the  same  connectivity  and  should  have  the  same  number  of 
boundaries. 

As  already  indicated,  this  equivalence  is  a  geometrical  equivalence : 
deformation  may  be  (but  is  not  of  necessity)  physically  possible. 

Similarly,  the  presence  of  one  or  of  several  knots  in  a  surface  makes  no 
essential  difference  in  the  use  of  the  surface  for  representing  a  variable.  Thus 
a  long  cylindrical  surface  is  changed  into  an  anchor-ring  when  its  ends  are 
joined  together ;  but  the  changed  surface  would  be  equally  effective  for 
purposes  of  representation  if  a  knot  were  tied  in  the  cylindrical  surface  before 
the  ends  are  joined. 

But  it  need  hardly  be  pointed  out  that  though  surfaces,  thus  twisted  or 
knotted,  are  equivalent  for  the  purpose  indicated,  they  are  not  equivalent  for 
all  topological  enumerations. 

Seeing  that  bifacial  surfaces,  with  the  same  connectivity  and  the  same 
number  of  boundaries,  are  equivalent  to  one  another,  it  is  natural  to  adopt,  as 
the  surface  of  reference,  some  simple  surface  with  those  characteristics;  thus 
for  a  surface  of  connectivity  2p  +  1  with  a  single  boundary,  the  surface  of  a 
solid  sphere,  bounded  by  a  point  and  pierced  through  with  p  holes,  could  be 
adopted. 

Klein  calls*  such  a  surface  of  reference  a  Normal  Surface. 

It  has  been  seen  that  a  bounded  spherical  surface  and  a  bounded  simply  connected 
part  of  a  plane  are  equivalent — they  are,  moreover,  physically  deformable  into  one 
another. 

An  untwisted  closed  ribbon  is  equivalent  to  a  bounded  piece  of  a  plane  with  one  hole 
in  it — they  are  deformable  into  one  another :  but  if  the  ribbon,  previous  to  being  closed, 
have  undergone  an  even  number  of  twists  each  through  180°,  they  are  still  equivalent 
but  are  not  physically  deformable  into  one  another.  Each  of  the  bifacial  surfaces  is 
doubly  connected  (for  a  single  cross-cut  renders  each  simply  connected)  and  each  of  them 

*  Ueber  Riemann's  Theorie  der  algebraischen  Functionen  und  ihrer  Integrate,  (Leipzig, 
Teubner,  1882),  p.  26. 


170.]  REFERENCES  335 

has  two  boundaries.  If  however  the  ribbon,  previous  to  being  closed,  have  imdcrgone 
an  odd  number  of  twists  each  through  180°,  the  surface  thus  obtained  is  not  equivalent  to 
the  single-holed  portion  of  the  plane  ;  it  is  unifacial  arid  has  only  one  boundary. 

A  spherical  surface  pierced  in  n-\-l  holes  is  equivalent  to  a  bounded  portion  of  the 
plane  with  n  holes  ;  each  is  of  connectivity  n  + 1  and  has  n  +  1  boundaries.  The  spherical 
surface  can  be  deformed  into  the  plane  surface  by  stretching  one  of  its  holes  into  the  form 
of  the  outside  boundary  of  the  plane  surface. 

Ex.  Prove  that  the  surface  of  a  bounded  anchor-ring  can  be  physically  deformed  into 
the  surface  in  Fig.  47,  p.  332. 


For  continuation  and  fuller  development  of  the  subjects  of  the  present  chapter,  the 
following  references,  in  addition  to  those  which  have  been  given,  will  be  found  useful  : 

Klein,  Math.  Ann.,  t.  vii,  (1874),  pp.  548—557;  ib.,  t.  ix,  (1876),  pp.  476—482. 

Lippich,   Math.  Ann.,  i.  vii,  (1874),  pp.   212 — 229  ;    Wiener  Sitzungsb.,   t.    Ixix,    (ii), 
(1874),  pp.  91—99. 

Durege,    Wiener  Sitzungsb.,  t.  Ixix,  (ii),  (1874),  pp.  115—120;   and  section  9  of  his 
treatise,  quoted  on  p.  316,  note. 

Neumann,  chapter  vii  of  his  treatise,  quoted  on  p.  5,  note. 

Dyck,  Math.  Ann.,  t.  xxxii,  (1888),  pp.  457—512,  ib.,  t.  xxxvii,  (1890),  pp.  273—316; 

at  the  beginning  of  the  first  part  of  this  investigation,  a  valuable  series  of  references 
is  given. 

Dingeldey,  Topologische  Studien,  (Leipzig,  Teubner,  1890). 


CHAPTER  XV. 

RIEMANN'S  SURFACES. 

171.  THE  method  of  representing  a  variable  by  assigning  to  it  a  position 
in  a  plane  or  on  a  sphere  is  effective  when  properties  of  uniform  functions  of 
that  variable  are  discussed.  But  when  multiform  functions,  or  integrals  of 
uniform  functions  occur,  the  method  is  effective  only  when  certain  parts  of 
the  plane  are  excluded,  due  account  being  subsequently  taken  of  the  effect  of 
such  exclusions;  and  this  process,  the  extension  of  Cauchy's  method,  was 
adopted  in  Chapter  IX. 

There  is  another  method,  referred  to  in  §  100  as  due  to  Riemann,  of  an 
entirely  different  character.  In  Riemann's  representation,  the  region,  in 
which  the  variable  z  exists,  no  longer  consists  of  a  single  plane  but  of  a 
number  of  planes ;  they  are  distinct  from  one  another  in  geometrical  concep 
tion,  yet,  in  order  to  preserve  a  representation  in  which  the  value  of  the 
variable  is  obvious  on  inspection,  the  planes  are  infinitesimally  close  to  one 
another.  The  number  of  planes,  often  called  sheets,  is  the  same  as  the 
number  of  distinct  values  (or  branches)  of  the  function  w  for  a  general 
argument  z  and,  unless  otherwise  stated,  will  be  assumed  finite;  each  sheet 
is  associated  with  one  branch  of  the  function,  and  changes  from  one  branch 
of  the  function  to  another  are  effected  by  making  the  ^-variable  change 
from  one  sheet  to  another,  so  that,  to  secure  the  possibility  of  change 
of  sheet,  it  is  necessary  to  have  means  of  passage  from  one  sheet  to  another. 
The  aggregate  of  all  the  sheets  is  a  surface,  often  called  a  Riemanns 
Surface. 

For  example,  consider  the  function 

w=z*  +  (z-I}~*, 

the  cube  roots  being  independent  of  one  another.     It  is  evidently  a  nine-valued  function  ; 
the  number  of  sheets  in  the  appropriate  Eiemann's  surface  is  therefore  nine. 

The  branch-points  are  2  =  0,  z  =  l,  2=00.  Let  o>  and  a  denote  a  cube-root  of  unity, 
independently  of  one  another  ;  then  the  values  of  z*  can  be  represented  in  the  form 


171.] 


EXAMPLES   OF   RIEMANN's   SURFACES 


337 


ill  -A  -  4 

23,    C023",    co22*;    and  the  values  of  (2-!)    3  can   be  represented  in   the  form  (2-!)      , 

^•(z  -  \ )  ~  3}  0  («  - 1)    »     The  nine  values  of  w  can  be  symbolically  expressed  as  follows  : — 


Fig.  49. 


Fig.  50. 


where  the  symbols  opposite  to  w  give  the  coefficients  of  z3  and  of  (2-  1)    3  respectively. 

Now  when  2  describes  a  small  simple  circuit  positively  round  the  origin,  the  groups 
in  cyclical  order  are  u\,  w2,  w3;  w4,  w5,  w6;  wr,  w8,  io9.  And  therefore,  in  the  immediate 
vicinity  of  the  origin,  there  must  be  means  of  passage  to  enable 
the  2-point  to  make  the  corresponding  changes  from  sheet  to  — 
sheet.  Taking  a  section  of  the  whole  surface  near  the  origin  ~ 
so  as  to  indicate  the  passages  and  regarding  the  right-hand 
sides  as  the  part  from  which  the  2-variable  moves  when  it  — 
describes  a  circuit  positively,  the  passages  must  be  in  character  as 
indicated  in  Fig.  49.  And  it  is  evident  that  the  further  descrip 
tion  of  small  simple  circuits  round  the  origin  will,  with  these  passages,  lead  to  the  proper 
values  :  thus  %,  which  after  the  single  description  is  the  value  of  w4,  becomes  w6  after 
another  description  and  it  is  evident  that  a  point  in  the  w-0  sheet  passes  into  the  w6  sheet. 

When  2  describes  a  small  simple  circuit  positively  round  the  point  1,  the  groups  in  cyclical 
order  are  wlt  ^4,  %;  w2,  w5,  ws;  w3,  w6,  w9:  and  therefore, 
in  the   immediate   vicinity   of  the   point  1,   there   must   be     ~ 
means  of  passage  to  render  possible  the  corresponding  changes 
of  2  from  sheet  to  sheet.     Taking  a  section  as  before  near  the     ~ 
point  1  and  with  similar  convention  as  to  the  positive  direc 
tion  of  the  2-path,  the  passages  must  be  in  character  as 
indicated  in  Fig.  50. 

Similarly  for  infinitely  large  values  of  2. 

If  then  the  sheets  can  be  so  joined  as  to  give  these  possibilities  of  passage  and  also 
give  combinations  of  them  corresponding  to  combinations  of  the  simple  paths  indicated, 
then  there  will  be  a  surface  to  any  point  of  which  will  correspond  one  and  only  one  value 
of  w  :  and  when  the  value  of  w  is  given  for  a  point  2  in  an  ordinary  plane  of  variation, 
then  that  value  of  w  will  determine  the  sheet  of  the  surface  in  which  the  point  2  is  to 
be  taken.  A  surface  will  then  have  been  constructed  such  that  the  function  w,  which  is 
multiform  for  the  single-plane  representation  of  the  variable,  is  uniform  for  variations 
in  the  many-sheeted  surface. 

Again,  for  the  simple  example  arising  from  the  two-valued  function,  defined  by 
the  equation 

w  =  {(z-a}(z-b}(z-c}}-\ 

the  branch-points  are  a,  b,  c,  oo  ;  and  a  small  simple  circuit  round  any  one  of  these 
four  points  interchanges  the  two  values.  The  Riemann's  surface  is  two-sheeted  and 
there  must  be  means  of  passage  between  the  two  sheets  in  the  vicinity  of  a,  that  of  b, 
that  of  c  and  at  the  infinite  part  of  the  plane. 

These  examples  are  sufficient  to  indicate  the  main  problem.     It  is  the 
construction  of  a  surface  in  which  the  independent  variable  can  move  so 
F.  22 


338  SHEETS  OF  HIEMANN'S  SURFACE  [171. 

that,  for  variations  of  z  in  that  surface,  the  multiformity  of  the  function  is 
changed  to  uniformity.  From  the  nature  of  the  case,  the  character  of  the 
surface  will  depend  on  the  character  of  the  function  :  and  thus,  though  all  the 
functions  are  uniform  within  their  appropriate  surfaces,  these  surfaces  are 
widely  various.  Evidently  for  uniform  functions  of  z  the  appropriate  surface 
on  the  above  method  is  the  single  plane  already  adopted. 

172.  The  simplest  classes  of  functions  for  which  a  Riemaim's  surface  is 
useful  are  (i)  those  called  (§  94)  algebraic  functions,  that  is,  multiform  functions 
of  the  independent  variable  denned  by  an  algebraical  equation  of  the  form 


which  is  of  finite  degree,  say  n,  in  w,  and  (ii)  those  usually  called  Abelian 
functions,  which  arise  through  integrals  connected  with  algebraic  functions. 

Of  such  an  algebraic  function  there  are,  in  general,  n  distinct  values  ;  but 
for  the  special  values  of  z,  that  are  the  branch-points,  two  or  more  of  the 
values  coincide.  The  appropriate  Riemann's  surface  is  composed  of  n  sheets  ; 
one  branch,  and  only  one  branch,  of  w  is  associated  with  a  sheet.  The 
variable  z,  in  its  relation  to  the  function,  is  determined  not  merely  by  its 
modulus  and  argument  but  also  by  its  sheet  ;  that  is,  in  the  language  of  the 
earlier  method,  we  take  account  of  the  path  by  which  z  acquires  a  value.  The 
particular  sheet  in  which  z  lies  determines  the  particular  branch  of  the 
function.  Variations  of  #,  which  occur  within  a  sheet  and  do  not  coincide 
with  points  lying  in  regions  of  passage  between  the  sheets,  lead  to  variations 
in  the  value  of  the  branch  of  w  associated  with  the  sheet  ;  a  return  to  an 
initial  value  of  z,  by  a  path  that  nowhere  lies  within  a  region  of  passage, 
leaves  the  ^-point  in  the  same  sheet  as  at  first  and  so  leads  to  the  initial 
branch  (and  to  the  initial  value  of  the  branch)  of  w.  But  a  return  to  an 
initial  value  of  z  by  a  path,  which,  in  the  former  method  of  representation, 
would  enclose  a  branch-point,  implies  a  change  of  the  branch  of  the  function 
according  to  the  definite  order  prescribed  by  the  branch-point.  Hence  the 
final  value  of  the  variable  z  on  the  Riemann's  surface  must  lie  in  a  sheet  that 
is  different  from  that  of  the  initial  (and  algebraically  equal)  value  ;  and 
therefore  the  sheets  must  be  so  connected  that,  in  the  immediate  vicinity  of 
branch-points,  there  are  means  of  passage  from  one  sheet  to  another,  securing 
the  proper  interchanges  of  the  branches  of  the  function  as  defined  by  the 
equation. 

173.  The  first  necessity  is  therefore  the  consideration  of  the  mode  in 
which  the  sheets  of  a  Riemann's  surface  are  joined  :  the  mode  is  indicated  by 
the  theorem  that  sheets  of  a  Riemann's  surface  are  joined  along  lines. 

The  junction  might  be  made  either  at  a  point,  as  with  two  spheres  in 
contact,  or  by  a  common  portion  of  a  surface,  as  with  one  prism  lying  on 


173.]  JOINED   ALONG    BRANCH-LINES  339 

another,  or  along  lines  ;  but  whatever  the  character  of  the  junction  be,  it 
must  be  such  that  a  single  passage  across  it  (thereby  implying  entrance  to 
the  junction  and  exit  from  it)  must  change  the  sheet  of  the  variable. 

If  the  junction  were  at  a  point,  then  the  £- variable  could  change  from  one 
sheet  into  another  sheet,  only  if  its  path  passed  through  that  point :  any 
other  closed  path  would  leave  the  z- variable  in  its  original  sheet.  A  small 
closed  curve,  infinites!  rn  ally  near  the  point  and  enclosing  it  and  no  other 
branch-point,  is  one  which  ought  to  transfer  the  variable  to  another  sheet 
because  it  encloses  a  branch-point :  and  this  is  impossible  with  a  point-junction 
when  the  path  does  not  pass  through  the  point.  Hence  a  junction  at  a  point 
only  is  insufficient  to  provide  the  proper  means  of  passage  from  sheet  to 
sheet. 

If  the  junction  were  effected   by  a  common  portion 
of    surface,    then    a   passage    through    it    (implying    an 
entrance  into  that  portion  and  an  exit  from  it)  ought  to 
change  the  sheet.     But,  in  such  a  case,  closed  contours          .-'--'' 
can  be  constructed  which  make  such  a  passage  without  Fi8-  51> 

enclosing  the  branch-point  a :  thus  the  junction  would  cause  a  change  of 
sheet  for  certain  circuits  the  description  of  which  ought  to  leave  the 
z- variable  in  the  original  sheet.  Hence  a  junction  by  a  continuous  area  of 
surface  does  not  provide  proper  means  of  passage  from  sheet  to  sheet. 

The  only  possible  junction  which  remains   is  a  line. 

The  objection  in  the  last  case  does  not  apply  to  a  closed      •  /  '^ 

contour  which  does  not  contain  the  branch-point ;  for  the  /.--"'' 

line   cuts  the   curve  twice   and   there  are  therefore   two  Fig.  52. 

crossings ;  the  second  of  them  makes  the  variable  return  to  the  sheet  which 
the  first  crossing  compelled  it  to  leave. 

Hence  the  junction  between  any  two  sheets  takes  place  along  a  line. 

Such  a  line  is  called*  a  branch-line.  The  branch -points  of  a  multiform 
function  lie  on  the  branch-lines,  after  the  foregoing  explanations ;  and  a 
branch-line  can  be  crossed  by  the  variable  only  if  the  variable  change  its 
sheet  at  crossing,  in  the  sequence  prescribed  by  the  branch-point  of  the 
function  which  lies  on  the  line.  Also,  the  sequence  is  reversed  when  the 
branch-line  is  crossed  in  the  reversed  direction. 

Thus,  if  two  sheets  of  a  surface  be  connected  along  a  branch-line,  a  point  which 
crosses  the  line  from  the  first  sheet  must  pass  into  the  second  and  a  point  which  crosses 
the  line  from  the  second  sheet  must  pass  into  the  first. 

Again,  if,  along  a  common  direction  of  branch-line,  the  first  sheet  of  a  surface 
be  connected  with  the  second,  the  second  with  the  third,  and  the  third  with 

*  Sometimes  cross-line,  sometimes  branch-section.  The  German  title  is  Verzweigungschnitt; 
the  French  is  lignc  de  passage ;  see  also  the  note  on  the  equivalents  of  branch-point,  p.  15. 

22—2 


340  PROPERTIES   OF   BRANCH-LINES  [173. 

the  first,  a  point  which  crosses  the  line  from  the  first  sheet  in  one  direction  must  pass 
into  the  second  sheet,  but  if  it  cross  the  line  in  the  other  direction  it  must  pass  into 
the  third  sheet. 

A  branch -point  does  not  necessarily  affect  all  the  branches  of  a  function : 
when  it  affects  only  some  of  them,  the  corresponding  property  of  the  Riemann's 
surface  is  in  evidence  as  follows.  Let  z=a  determine  a  branch-point  affecting, 
say,  only  r  branches.  Take  n  points  a,  one  in  each  of  the  sheets  ;  and  through 
them  draw  n  lines  cab,  having  the  same  geometrical  position  in  the  respective 
sheets.  Then  in  the  vicinity  of  the  point  a  in  each  of  the  n  sheets,  associated 
with  the  r  affected  branches,  there  must  be  means  of  passage  from  each  one 
to  all  the  rest  of  them ;  and  the  lines  cab  can  conceivably  be  the  branch-lines 
with  a  properly  established  sequence.  The  point  a  does  not  affect  the  other 
n  —  r  branches :  there  is  therefore  no  necessity  for  means  of  passage  in  the 
vicinity  of  a  among  the  remaining  n  —  r  sheets.  In  each  of  these  remaining 
sheets,  the  point  a  and  the  line  cab  belong  to  their  respective  sheets  alone : 
for  them,  the  point  a  is  not  a  branch-point  and  the  line  cab  is  not  a  branch- 
line. 

174.  Several  essential  properties  of  the  branch-lines  are  immediate 
inferences  from  these  conditions. 

I.  A  free  end  of  a  branch-line  in  a  surface  is  a  branch-point. 

Let  a  simple  circuit  be  drawn  round  the  free  end  so  small  as  to  enclose  no 
branch-point  (except  the  free  end,  if  it  be  a  branch-point).  The  circuit  meets 
the  branch-line  once,  and  the  sheet  is  changed  because  the  branch-line  is 
crossed ;  hence  the  circuit  includes  a  branch-point  which  therefore  can  be 
only  the  free  end  of  the  line. 

Note.  A  branch-line  may  terminate  in  the  boundary  of  the  surface, 
and  then  the  extremity  need  not  be  a  branch-point. 

II.  When  a  branch-line  extends  beyond  a  branch-point  lying  in  its  course, 
the  sequence  of  interchange  is  not  the  same  on  the  two  sides  of  the  point. 

If  the  sequence  of  interchange  be  the  same  on  the  two  sides  of  the  branch 
point,  a  small  circuit  round  the  point  would  first  cross  one  part  of  the  branch- 
line  and  therefore  involve  a  change  of  sheet  and  then,  in  its  course,  would 
cross  the  other  part  of  the  branch-line  in  the  other  direction  which,  on  the 
supposition  of  unaltered  sequence,  would  cause  a  return  to  the  initial  sheet. 
In  that  case,  a  circuit  round  the  branch-point  would  fail  to  secure  the  proper 
change  of  sheet.  Hence  the  sequence  of  interchange  caused  by  the  branch- 
line  cannot  be  the  same  on  the  two  sides  of  the  point. 

III.  If  two  branch-lines  with  different  sequences  of  interchange  have  a 
common  extremity,  that  point  is  either  a  branch-point  or  an  extremity  of  at 
least  one  other  branch-line. 


174.]  SYSTEM   OF   BRANCH-LINES  341 

If  the  point  be  not  a  branch-point,  then  a  simple  curve  enclosing  it,  taken 
so  small  as  to  include  no  branch-point,  must 
leave  the  variable  in  its  initial  sheet.     Let  A 
be  such  a  point,  AB  and  AC  be  two  branch- 
lines  having  A  for  a  common  extremity ;  let  .,  A  ,.•        — ^ « 

the  sequence  be  as  in  the  figure,  taken  for  a  F. 

simple  case ;   and  suppose  that  the  variable 

initially  is  in  the  rth  sheet.  A  passage  across  AB  makes  the  variable 
pass  into  the  sth  sheet.  If  there  be  no  branch-line  between  AB  and  AC 
having  an  extremity  at  A,  and  if  neither  n  nor  m  be  s,  then  the  passage 
across  AC  makes  no  change  in  the  sheet  of  the  variable  and,  therefore,  in 
order  to  restore  r  before  AB,  at  least  one  branch-line  must  lie  in  the  angle 
between  AC  and  AB,  estimated  in  the  positive  trigonometrical  sense. 

If  either  n  or  m,  say  n,  be  s,  then  after  passage  across  AC,  the  point  is  in 
the  mt\i  sheet ;  then,  since  the  sequences  are  not  the  same,  m  is  not  r  and 
there  must  be  some  branch-line  between  AC  and  AB  to  make  the  point 
return  to  the  rth  sheet  on  the  completion  of  the  circuit. 

If  then  the  point  A  be  not  a  branch-point,  there  must  be  at  least  one 
other  branch-line  having  its  extremity  at  A.  This  proves  the  proposition. 

COROLLARY  1.  If  both  of  two  branch-lines  extend  beyond  a  point  of  inter 
section,  which  is  not  a  branch-point,  no  sheet  of  the  surface  has  both  of  them  for 
branch-lines. 

COROLLARY  2.  If  a  change  of  sequence  occur  at  any  point  of  a  branch- 
line,  then  either  that  point  is  a  branch-point  or  it  lies  also  on  some  other 
branch-line. 

COROLLARY  3.  No  part  of  a  branch-line  with  only  one  branch-point  on  it 
can  be  a  closed  curve. 

It  is  evidently  superfluous  to  have  a  branch-line  without  any  branch-point 
on  it. 

175.  On  the  basis  of  these  properties,  we  can  obtain  a  system  of  branch- 
lines  satisfying  the  requisite  conditions  which  are  : — 

(i)  the  proper  sequences  of  change  from  sheet  to  sheet  must  be 
secured  by  a  description  of  a  simple  circuit  round  a  branch 
point  :  if  this  be  satisfied  for  each  of  the  branch-points,  it 
will  evidently  be  satisfied  for  any  combination  of  simple  circuits, 
that  is,  for  any  path  whatever  enclosing  one  or  more  branch 
points. 

(ii)  the  sheet,  in  which  the  variable  re-assumes  its  initial  value  after 
describing  a  circuit  that  encloses  no  branch-point,  must  be  the 
initial  sheet. 


342  SYSTEM   OF   BRANCH-LINES  [175. 

In  the  ^-plane  of  Cauchy's  method,  let  lines  be  drawn  from  any  point  I,  not 
a  branch-point  in  the  first  instance,  to  each  of  the  branch-points,  as  in  fig.  19, 
p.  156,  so  that  the  joining  lines  do  not  meet  except  at  /:  and  suppose  the 
w-sheeted  Riemann's  surface  to  have  branch-lines  coinciding  geometrically 
with  these  lines,  as  in  §  173,  and  having  the  sequence  of  interchange  for 
passage  across  each  the  same  as  the  order  in  the  cycle  of  functional  values 
for  a  small  circuit  round  the  branch-point  at  its  free  end.  No  line  (or  part 
of  a  line)  can  be  a  closed  curve ;  the  lines  need  not  be  straight,  but  they 
will  be  supposed  drawn  as  direct  as  possible  to  the  points  in  angular 
succession. 

The  first  of  the  above  requisite  conditions  is  satisfied  by  the  establish 
ment  of  the  sequence  of  interchange. 

To  consider  the  second  of  the  conditions,  it  is  convenient  to  divide 
circuits  into  two  kinds,  (a)  those  which  exclude  /,  (/3)  those  which  include  /, 
no  one  of  either  kind  (for  our  present  purpose)  including  a  branch-point. 

A  closed  circuit,  excluding  I  and  all  the  branch-points,  must  intersect  a 
branch-line  an  even  number  of  times,  if 
it  intersect  the  line  in  real  points.  Let 
the  figure  (fig.  54)  represent  such  a  case : 
then  the  crossings  at  A  and  B  counter 
act  one  another  and  so  the  part  be 
tween  A  and  B  may  without  effect  be 
transferred  across  IB3  so  as  not  to  cut 
the  branch-line  at  all.  Similarly  for 
the  points  C  and  D :  and  a  similar 
transference  of  the  part  now  between 
C  and  D  may  be  made  across  the 
branch-line  without  effect:  that  is,  the 
circuit  can,  without  effect,  be  changed 
so  as  not  to  cut  the  branch-line  IBS  at  all.  A  similar  change  can  be  made 
for  each  of  the  branch-lines :  and  so  the  circuit  can,  without  effect,  be  changed 
into  one  which  meets  no  branch-line  and  therefore,  on  its  completion,  leaves 
the  sheet  unchanged. 

A  closed  circuit,  including  /  but  no  branch-point,  must  meet  each  branch- 
line  an  odd  number  of  times.  A  change  similar  in  character  to  that  in 
the  previous  case  may  be  made  for  each  branch-line  :  and  without  affecting 
the  result,  the  circuit  can  be  changed  so  that  it  meets  each  branch-line  only 
once.  Now  the  effect  produced  by  a  branch-line  on  the  function  is  the  same 
as  the  description  of  a  simple  loop  round  the  branch-point  which  with  / 
determines  the  branch-line :  and  therefore  the  effect  of  the  circuit  at  present 
contemplated  is,  after  the  transformation  which  does  not  affect  the  result,  the 
same  as  that  of  a  circuit,  in  the  previously  adopted  mode  of  representation, 


175.]  FOR   A   SURFACE 

enclosing  all  the  branch-points.  But,  by  Cor.  III.  of  §  90,  the  effect  of  a 
circuit  which  encloses  all  the  branch-points  (including  z  =  GO  ,  if  it  be  a 
branch-point)  is  to  restore  the  value  of  the  function  which  it  had  at  the 
beginning  of  the  circuit :  and  therefore  in  the  present  case  the  effect  is  to 
make  the  point  return  to  the  sheet  in  which  it  lay  initially. 

It  follows  therefore  that,  for  both  kinds  of  a  closed  circuit  containing  no 
branch-point,  the  effect  is  to  make  the  ^-variable  return  to  its  initial  sheet 
on  resuming  its  initial  value  at  the  close  of  the  circuit. 

Next,  let  the  point  /  be  a  branch-point ;  and  let  it  be  joined  by  lines, 
as  direct*  as  possible,  to  each  of  the  other  branch -points  in  angular  succes 
sion.  These  lines  will  be  regarded  as  the  branch-lines ;  and  the  sequence  of 
interchange  for  passage  across  any  one  is  made  that  of  the  interchange  pre 
scribed  by  the  branch-point  at  its  free  extremity. 

The  proper  sequence  of  change  is  secured  for  a  description  of  a  simple 
closed  circuit  round  each  of  the  branch-points  other  than  /.  Let  a  small 
circuit  be  described  round  /;  it  meets  each  of  the  branch-lines  once  and 
therefore  its  effect  is  the  same  as,  in  the  language  of  the  earlier  method  of 
representing  variation  of  z,  that  of  a  circuit  enclosing  all  the  branch-points 
except  7.  Such  a  circuit,  when  taken  on  the  Neumann's  sphere,  as  in  Cor. 
III.,  §  90  and  Ex.  2,  §  104,  may  be  regarded  in  two  ways,  according  as  one  or 
other  of  the  portions,  into  which  it  divides  the  area  of  the  sphere,  is  regarded 
as  the  included  area;  in  one  way,  it  is  a  circuit  enclosing  all  the  branch 
points  except  /,  in  the  other  it  is  a  circuit  enclosing  /  alone  and  no  other 
branch-point.  Without  making  any  modification  in  the  final  value  of  w,  it 
can  (by  §  90)  be  deformed,  either  into  a  succession  of  loops  round  all  the 
branch-points  save  one,  or  into  a  loop  round  that  one ;  the  effect  of  these  two 
deformations  is  therefore  the  same.  Hence  the  effect  of  the  small  closed 
circuit  round  /  meeting  all  the  branch-lines  is  the  same  as,  in  the  other  mode 
of  representation,  that  of  a  small  curve  round  /  enclosing  no  other  branch 
point  ;  and  therefore  the  adopted  set  of  branch- lines  secures  the  proper 
sequence  of  change  of  value  for  description  of  a  circuit  round  /. 

The  first  of  the  two  necessary  conditions  is  therefore  satisfied  by  the 
present  arrangement  of  branch-lines. 

The  proof,  that  the  second  of  the  two  necessary  conditions  is  also  satisfied 
by  the  present  arrangement  of  branch-lines,  is  similar  to  that  in  the  preceding 
case,  save  that  only  the  first  kind  of  circuit  of  the  earlier  proof  is  possible. 

Jt  thus  appears  that  a  system  of  branch-lines  can  be  obtained  which 
secures  the  proper  changes  of  sheet  for  a  multiform  function  :  and  therefore 
Riemann's  surfaces  can  be  constructed  for  such  a  function,  the  essential 
property  being  that  over  its  appropriate  surface  an  otherwise  multiform 
function  of  the  variable  is  a  uniform  function. 

*  The  reason  for  this  will  appear  in  §§  183,  184. 


344  EXAMPLES  [175. 

The  multipartite  character  of  the  function  has  its  influence  preserved  by 
the  character  of  the  surface  to  which  the  function  is  referred  :  the  surface, 
consisting  of  a  number  of  sheets  joined  to  one  another,  may  be  a  multiply 
connected  surface. 

In  thus  proving  the  general  existence  of  appropriate  surfaces,  there  has 
remained  a  large  arbitrary  element  in  their  actual  construction  :  moreover, 
in  particular  cases,  there  are  methods  of  obtaining  varied  configurations  of 
branch-lines.  Thus  the  assignment  of  the  n  branches  to  the  n  sheets  has 
been  left  unspecified,  and  is  therefore  so  far  arbitrary  :  the  point  I,  if  not  a 
branch-point,  is  arbitrarily  chosen  and  so  there  is  a  certain  arbitrariness  of 
position  in  the  branch  -lines.  Naturally,  what  is  desired  is  the  simplest 
appropriate  surface  :  the  particularisation  of  the  preceding  arbitrary  qualities 
is  used  to  derive  a  canonical  form  of  the  surface. 

176.  The  discussion  of  one  or  two  simple  cases  will  help  to  illustrate  the 
mode  of  junction  between  the  sheets,  made  by  branch-lines. 

The  simplest  case  of  all  is  that  in  which  the  surface  has  only  a  single 
sheet:  it  does  not  require  discussion. 

The  case  next  in  simplicity  is  that  in  which  the  surface  is  two-sheeted  : 
the  function  is  therefore  two-  valued  and  is  consequently  defined  by  a 
quadratic  equation  of  the  form 

Lua  +  2Mu  +  N  =  0, 

where  L  and  M  are  uniform  functions  of  z.  When  a  new  variable  w  is 
introduced,  defined  by  Lu  +  M=w,  so  that  values  of  iv  and  of  u  correspond 
uniquely,  the  equation  is 


It  is  evident  that  every  branch-point  of  u  is  a  branch-point  of  w,  and 
vice  versa  ;  hence  the  Riemann's  surface  is  the  same  for  the  two  equations. 
Now  any  root  of  P  (z)  of  odd  degree  is  a  branch-point  of  iv.  If  then 


where  R  (z}  is  a  product  of  only  simple  factors,  every  factor  of  R  (z)  leads  to 
a  branch-point.  If  the  degree  of  R  (z}  be  even,  the  number  of  branch-points 
for  finite  values  of  the  variable  is  even  and  z  =  oo  is  not  a  branch-point  ;  if  the 
degree  of  R(z)  be  odd,  the  number  of  branch  -points  for  finite  values  of  the 
variable  is  odd  and  z  =  oo  is  a  branch-point  :  in  either  case,  the  number  of 
branch-points  is  even. 

There  are  only  two  values  of  w,  and  the  Riemann's  surface  is  two-sheeted: 
crossing  a  branch-line  therefore  merely  causes  a  change  of  sheet.  The  free 
ends  of  branch-lines  are  branch-points  ;  a  small  circuit  round  any  branch 
point  causes  an  interchange  of  the  branches  w,  and  a  circuit  round  any  two 
branch-points  restores  the  initial  value  of  w  at  the  end  and  therefore  leaves 
the  variable  in  the  same  sheet  as  at  the  beginning.  These  are  the  essential 
requirements  in  the  present  case  ;  all  of  them  are  satisfied  by  taking  each 


176.]  OF  RIEMANN'S  SURFACES  345 

branch-line  as  a  line  connecting  two  (and  only  two)  of  the  branch-points.  The 
ends  of  all  the  branch  -lines  are  free  :  and  their  number,  in  this  method,  is 
one-half  that  of  the  (even)  number  of  branch-points.  A  small  circuit  round 
a  branch-point  meets  a  branch-line  once  and  causes  a  change  of  sheet  ;  a 
circuit  round  two  (and  not  more  than  two)  branch  -points  causes  either  no 
crossing  of  branch-line  or  an  even  number  of  crossings  and  therefore  restores 
the  variable  to  the  initial  sheet. 

A  branch-line  is,  in  this  case,  usually  drawn  in  the  form  of  a  straight  line 
when  the  surface  is  plane  :  but  this  form  is  not  essential  and  all  that  is 
desirable  is  to  prevent  intersections  of  the  branch-lines. 

Note.  Junction  between  the  sheets  along  a  branch-line  is  easily  secured. 
The  two  sheets  to  be  joined  are  cut  along  the  branch-line.  One  edge  of  the 
cut  in  the  upper  sheet,  say  its  right  edge  looking  along  the  section,  is  joined 
to  the  left  edge  of  the  cut  in  the  lower  sheet  ;  and  the  left  edge  in  the  upper 
sheet  is  joined  to  the  right  edge  in  the  lower. 

A  few  simple  examples  will  illustrate  these  remarks  as  to  the  sheets  :  illustrations  of 
closed  circuits  will  arise  later,  in  the  consideration  of  integrals  of  multiform  functions. 

Ex.  1.     Let  w*  =  A(z-a)(z-b}, 

so  that  a  and  b  are  the  only  branch-points.  The  surface  is  two-sheeted  :  the  line  ab  may 
be  made  the  branch-line.  In  Fig.  55  only  part  of  the  upper  sheet  is  shewn*,  as  likewise 
only  part  of  the  lower  sheet.  Continuous  lines  imply  what  is  visible  ;  arid  dotted  lines 
what  is  invisible,  on  the  supposition  that  the  sheets  are  opaque. 

The  circuit,  closed  in  the  surface  and  passing  round  0,  is  made  up  of  OJK  in  the  upper 
sheet  :  the  point  crosses  the  branch-line  and  then  passes  into  the  lower  sheet,  where  it 
describes  the  dotted  line  KLH  :  it  then  meets  and  crosses  the  branch-line  at  If,  passes 
into  the  upper  sheet  and  in  that  sheet  returns  to  0.  Similarly  of  the  line  ABC,  the  part 
AB  lies  in  the  lower  sheet,  the  part  EC  in  the  upper  :  of  the  line  DG  the  part  DE  lies  in 
the  upper  sheet,  the  part  EFG  in  the  lower,  the  piece  FG  of  this  part  being  there  visible 
beyond  the  boundary  of  the  retained  portion  of  the  upper  surface. 

Ex.  2.     Let  Aw?2  =  z3-a3. 

The  branch-points  (Fig.  56)  are  A  (  =  a),  B  (  =  ««),  (7(  =  aa2),  where  a  is  a  primitive  cube 
root  of  unity,  and  2  =  00.  The  branch  -lines  can  be  made  by  BC,  Ace  ;  and  the  two- 
sheeted  surface  will  be  a  surface  over  which  w  is  uniform.  Only  a  part  of  each  sheet 
is  shewn  in  the  figure;  a  section  also  is  made  at  M  across  the  surface,  cutting  the  branch  - 
line  A  QO  . 

Ex.  3.     Let  wm=zn, 

where  n  and  TO  are  prime  to  each  other.  The  branch-points  are  z  =  0  and  2=00  ;  and  the 
branch-line  extends  from  0  to  QO  .  There  are  m  sheets  ;  if  we  associate  them  in  order  with 
the  branches  ws,  where 


wa=re 

for  s=l,  2,  ...,  TO,  then  the  first  sheet  is  connected  with  the  second  forwards,  the  second 
with  the  third  forwards,  and  so  on  ;  the  mth  being  connected  with  the  first  forwards. 

*  The  form  of  the  three  figures  in  the  plate  opposite  p.  346  is  suggested  by  Holzmiiller,  Ein- 
fiihrung  in  die  Theorie  der  isogonalen  Vericandschaften  und  der  confomien  AbbUdimgen,  (Leipzig, 
Teubner,  1882),  in  which  several  illustrations  are  given. 


346 


SPHERICAL  RIEMANN'S  SURFACE 


[176. 


The  surface  is  sometimes  also  called  a  winding-surface;  and  a  branch-point  such  as 
z—0  on  the  surface,  where  a  number  m  of  sheets  pass  into  one  another  in  succession,  is 
also  called  a  winding-point  of  order  m—  1  (see  p.  15,  note).  An  illustration  of  the  surface 
for  m  =  3  is  given  in  Fig.  57,  the  branch-line  being  cut  so  as  to  shew  the  branching  :  what 
is  visible  is  indicated  by  continuous  lines  ;  what  is  in  the  second  sheet,  but  is  invisible,  is 
indicated  by  the  thickly  dotted  line ;  what  is  in  the  third  sheet,  but  is  invisible,  is  indic 
ated  by  the  thinly  dotted  line. 

Ex.  4.  Consider  a  three-sheeted  surface  having  four  branch-points  at  a,  b,  c,  d ;  and 
let  each  point  interchange  two  branches,  say,  w.2,  w3  at  a  ;  iv^  w3ai  b ;  w2,  w3  at  c ;  wlt  w2 


at  d  ;    the  points  being  as  in  Fig.  58.     It  is  easy  to  verify  that  these   branch-points 
satisfy  the  condition  that  a  circuit,  enclosing  them  all,  restores  the  initial  value  of  w. 

The  branching  of  the  sheets  may  be  made  as  in  the  figure,  the  integers  on  the  two  sides 
of  the  line  indicating  the  sheets  that  are  to  be  joined  along  the  line. 

A  canonical  form  for  such  a  surface  can  be  derived  from  the  more  general  case  given 
later  (in  §§  186—189). 

Ex.  5.     Shew  that,  if  the  equation 


be  of  degree  n  in  w  and  be  irreducible,  all  the  n  sheets  of  the  surface  are  connected,  that 
is,  it  is  possible  by  an  appropriate  path  to  pass  from  any  sheet  to  any  other  sheet. 

177.  It  is  not  necessary  to  limit  the  surface  representing  the  variable  to 
a  set  of  planes;  and,  indeed,  as  with  uniform  functions,  there  is  a  convenience 
in  using  the  sphere  for  the  purpose. 

We  take  n  spheres,  each  of  diameter  unity,  touching  the  Riemann's  plane 
surface  at  a  point  A  ;  each  sphere  is  regarded  as  the  stereographic  projection 
of  a  plane  sheet,  with  regard  to  the  other  extremity  A'  of  the  spherical 
diameter  through  A.  Then,  the  sequence  of  these  spherical  sheets  being 
the  same  as  the  sequence  of  the  plane  sheets,  branch-points  in  the  plane 
surface  project  into  branch-points  on  the  spherical  surface  :  branch  -lines  be 
tween  the  plane  sheets  project  into  branch-lines  between  the  spherical  sheets 
and  are  terminated  by  corresponding  points  ;  and  if  a  branch-line  extend  in 
the  plane  surface  to  z=co,  the  corresponding  branch-line  in  the  spherical 
surface  is  terminated  at  A'. 

A  surface  will  thus  be  obtained  consisting  of  n  spherical  sheets;  like 
the  plane  Riemann's  surface,  it  is  one  over  which  the  n-valued  function  is  a 
uniform  function  of  the  position  of  the  variable  point. 


Fig. 


M  — =-00 


To  face  p.  346. 


Fig.  57. 


177.]  CONNECTIVITY   OF   A    RIEMANN's   SURFACE  347 

But  also  the  connectivity  of  the  n-sheeted  spherical  surface  is  the  same  as 
that  of  the  n-sheeted  plane  surface  with  which  it  is  associated. 

In  fact,  the  plane  surface  can  be  mechanically  changed  into  the  spherical 
surface  without  tearing,  or  repairing,  or  any  change  except  bending  and 
compression:  all  that  needs  to  be  done  is  that  the  n  plane  sheets  shall  be 
bent,  without  making  any  change  in  their  sequence,  each  into  a  spherical 
form,  and  that  the  boundaries  at  infinity  (if  any)  in  the  plane-sheet  shall 
be  compressed  into  an  infinitesimal  point,  being  the  South  pole  of  the  cor 
responding  spherical  sheet  or  sheets.  Any  junctions  between  the  plane 
sheets  extending  to  infinity  are  junctions  terminated  at  the  South  pole.  As 
the  plane  surface  has  a  boundary,  which,  if  at  infinity  on  one  of  the  sheets,  is 
therefore  not  a  branch-line  for  that  sheet,  so  the  spherical  surface  has  a 
boundary  which,  if  at  the  South  pole,  cannot  be  the  extremity  of  a  branch- 
line. 

178.  We  proceed  to  obtain  the  connectivity  of  a  Riemann's  surface :  it 
is  determined  by  the  following  theorem  : — 

Let  the  total  number  of  branch-points  in  a  Riemann's  n-sheeted  surface  be 
r ;  and  let  the  number  of,  branches  of  the  function  interchanging  at  the  first 
point  be  ml,  the  number  interchanging  at  the  second  be  m.2,  and  so  on.  Then 
the  connectivity  of  the  surface  is 

fl-2n  +  3, 
where  fl  denotes  m,1  +  m2  +  ...  +  mr  —  r. 

Take*  the  surface  in  the  bounded  spherical  form,  the  connectivity  N  of 
which  is  the  same  as  that  of  the  plane  surface :  and  let  the  boundary  be  a 
small  hole  A  in  the  outer  sheet.  By  means  of  cross-cuts  and  loop-cuts,  the 
surface  can  be  resolved  into  a  number  of  distinct  simply  connected  pieces. 

First,  make  a  slice  bodily  through  the  sphere,  the  edge  in  the 
outside  sheet  meeting  A  and  the  direction  of  the 
slice  through  A  being  chosen  so  that  none  of  the 
branch -points  lie  in  any  of  the  pieces  cut  off.  Then  n 
parts,  one  from  each  sheet  and  each  simply  connected, 
are  taken  away.  The  remainder  of  the  surface  has  a 
cup-like  form ;  let  the  connectivity  of  this  remainder 
be  M. 

This  slice  has  implied  a  number  of  cuts. 

The  cut  made  in  the  outside  sheet  is  a  cross-cut, 
because  it  begins  and  ends  in  the  boundary  A.  It 
divides  the  surface  into  two  distinct  pieces,  one  being 
the  portion  of  the  outside  sheet  cut  off,  and  this  piece  is  simply  connected ; 

*  The  proof  is  founded  on  Neumann's,  pp.  108 — 172. 


348  CONNECTIVITY   OF   A   SURFACE  [178. 

hence,  by  Prop.  III.  of  §  160,  the  remainder  has  its  connectivity  still  repre 
sented  by  N. 

The  cuts  in  all  the  other  sheets,  caused  by  the  slice,  are  all  loop-cuts, 
because  they  do  not  anywhere  meet  the  boundary.  There  are  n  —  1  loop- 
cuts,  and  each  cuts  off  a  simply  connected  piece ;  and  the  remaining  surface 
is  of  connectivity  M.  Hence,  by  Prop.  V.  of  §  161, 

M  +  n  -  1  =  N  +  2  (n  -  1), 
and  therefore  M  =  N+n—l. 

In  this  remainder,  of  connectivity  M,  make  r  —  1  cuts,  each  of  which 
begins  in  the  rim  and  returns  to  the  rim,  and  is  to  be  made  through  the  n 
sheets  together ;  and  choose  the  directions  of  these  cuts  so  that  each  of  the 
r  resulting  portions  of  the  surface  contains  one  (and  only  one)  of  the  branch 
points. 

Consider  the  portion  of  the  surface  which  contains  the  branch-point 
where  ml  sheets  of  the  surface  are  connected.  The  ml  connected  sheets 
constitute  a  piece  of  a  winding-surface  round  the  winding-point  of  order 
ml  —  1  ;  the  remaining  sheets  are  unaffected  by  the  winding-point,  and 
therefore  the  parts  of  them  are  n  —  m^  distinct  simply  connected  pieces. 
The  piece  of  winding- surface  is  simply  connected ;  because  a  circuit,  that 
does  not  contain  the  winding-point,  is  reducible  without  passing  over  the 
winding-point,  and  a  circuit,  that  does  contain  the  winding-point,  is  reducible 
to  the  winding-point,  so  that  no  irreducible  circuit  can  be  drawn.  Hence 
the  portion  of  the  surface  under  consideration  consists  of  n  —  ml  +  1  distinct 
simply  connected  pieces. 

Similarly  for  the  other  portions.  Hence  the  total  number  of  distinct 
simply  connected  pieces  is 

r 

2  (n  -  mq  +  1) 

9  =  1 

r 

=  m —  2  mq  +  r 
l-i 

=  nr  —  fl. 

But  in  the  portion  of  connectivity  M  each  of  the  r  —  1  cuts  causes,  in 
each  of  the  sheets,  a  cut  passing  from  the  boundary  and  returning  to  the 
boundary,  that  is,  a  cross-cut.  Hence  there  are  n  cross-cuts  from  each  of  the 
r—\  cuts,  and  therefore  n  (r—  1)  cross-cuts  altogether,  made  in  the  portion  of 
surface  of  connectivity  M. 

The  effect  of  these  n(r  —  1)  cross-cuts  is  to  resolve  the  portion  of  con 
nectivity  M  into  nr  —  £l  distinct  simply  connected  pieces  ;  hence,  by  §  160, 

M  =  n  (r  -  1)  -  (nr  -  H)  +  2, 

and  therefore  N  =  M  —  (n  —  1)  =  n  -  2n  +  3, 

the  connectivity  of  the  Riemann's  surface. 


178.]  CLASS   OF   A   SURFACE  349 

r 

The  quantity  H,  having  the  value   2  (mq  —  1),  may  be  called  the  rami- 

</=i 

fication  of  the   surface,  as  indicating  the  aggregate  sum  of  the   orders  of 
the  different  branch-points. 

Note.  The  surface  just  considered  is  a  closed  surface  to  which  a  point 
has  been  assigned  for  boundary;  hence,  by  Cor.  I.,  Prop.  III.,  §  164,  its 
connectivity  is  an  odd  integer.  Let  it  be  denoted  by  2p  +  1  ;  then 

2p  =  ft  -  2/i  +  2, 

and  2p  is  the  number  of  cross-cuts  which  change  the  Riemann's  surface  into 
one  that  is  simply  connected. 

The  integer  p  is  often  called  (Cor.  I.,  Prop.  III.,  §  164)  the  class  of  the 
Riemann's  surface;  and  the  equation 

f(w,  z)  =  0 

is  said  to  be  of  class  p,  when  p  is  the  class  of  the  associated  Riemann's 
surface. 

Ex.  1.     When  the  equation  is 

w>  =  \(z-a}(z-b\ 

we  have  a  two-sheeted  surface,  ?t  =  2.  There  are  two  branch-  points,  z  =  a  and  z  =  b;  but 
2=00  is  not  a  branch-point  ;  so  that  r=2.  At  each  of  the  branch-points  the  two  values  are 
interchanged,  so  that  m1  =  2,  ??i2  =  2;  thus  Q  =  2.  Hence  the  connectivity  =2-4  +  3  =  1, 
that  is,  the  surface  is  simply  connected. 

The  surface  can  be  deformed,  as  in  the  example  in  §  169,  into  a  sphere. 
Ex.  2.     When  the  equation  is 


we  have  ?t  =  2.  There  are  four  branch-points,  viz.,  et,  e2,  e3,  oc  ,  so  that  r  =  4  ;  and  at  each 
of  them  the  two  values  of  w  are  interchanged,  so  that  mg  =  2  (for  5  =  1,2,  3,  4),  and  therefore 
Q  =  8-  4  =  4.  Hence  the  connectivity  is  4-  4  +  3,  that  is,  3  ;  and  the  value  of  p  is  unity. 

Similarly,  the  surface  associated  with  the  equation 


where  U(z]  is  a  rational,  integral,  algebraical  function  of  degree  2«i  -  1  or  of  degree  2»i, 
is  of  connectivity  2wi  +  l  ;  so  that  p  =  m.     The  equation 

W2==(1_22)(1_^2) 

is  of  class  p=\.    The  case  next  in  importance  is  that  of  the  algebraical  equation  leading  to 
the  hyperelliptic  functions,  when  (/"is  either  a  quintic  or  a  sextic  ;  and  then  p  =  2. 

Ex.  3.     Obtain  the  connectivity  of  the  Riemann's  surface  associated  with  the  equation 

w3  +  ^  —  3awz  =  1  , 
where  a  is  a  constant,  (i)  when  a  is  zero,  (ii)  when  a  is  different  from  zero. 


•350  RESOLUTION    OF   A    RIEMANN's   SURFACE  [178. 

Ex.  4.     Shew  that,  if  the  surface  associated  with  the  equation 

f(w,z)  =  0, 

have  p.  boundary-lines  instead  of  one,  and  if  the  equation  have  the  same  branch-points 
as  in  the  foregoing  proposition,  the  connectivity  is  Q- 


179.  The  consideration  of  irreducible  circuits  on  the  surface  at  once 
reveals  the  multiple  connection  of  the  surface,  the  numerical  measure  of 
which  has  been  obtained.  In  a  Riemann's  surface,  a  simple 
closed  circuit  cannot  be  deformed  over  a  branch-point.  Let 
A  be  a  branch-point,  and  let  AE...  be  the  branch-line 
having  a  free  end  at  A.  Take  a  curve  ...CED...  crossing 
the  branch-line  at  E  and  passing  into  a  sheet  different 
from  that  which  contains  the  portion  CE  ;  and,  if  possible, 
let  a  slight  deformation  of  the  curve  be  made  so  as  to  transfer  the  portion 
CE  across  the  branch-point  A.  In  the  deformed  position,  the  curve 
...C'E'D'  '...  does  not  meet  the  branch-line;  there  is,  consequently,  no 
change  of  sheet  in  its  course  near  A  and  therefore  E'D'...,  which  is  the 
continuation  of  ...C'E',  cannot  be  regarded  as  the  deformed  position  of  ED. 
The  two  paths  are  essentially  distinct  ;  and  thus  the  original  path  cannot  be 
deformed  over  the  branch-point. 

It  therefore  follows  that  continuous  deformation  of  a  circuit  over  a 
branch-point  on  a  Riemann's  surface  is  a  geometrical  impossibility. 

Ex.  Trace  the  variation  of  the  curve  CED,  as  the  point  E  moves  up  to  A  and  then 
returns  along  the  other  side  of  the  branch-line. 

Hence  a  circuit  containing  two  or  more  of  the  branch-points  is  irreducible  ; 
but  a  circuit  containing  all  the  branch-points  is  equivalent  to  a  circuit  that 
contains  none  of  them,  and  it  is  therefore  reducible. 

If  a  circuit  contain  only  one  branch-point,  it  can  be  continuously  deformed 
so  as  to  coincide  with  the  point  on  each  sheet  and  therefore,  being  deformable 
into  a  point,  it  is  a  reducible  circuit.  An  illustration  has  already  occurred  in 
the  case  of  a  portion  of  winding-surface  containing  a  single  winding-point 
(p.  348);  all  circuits  drawn  on  it  are  reducible. 

It  follows  from  the  preceding  results  that  the  Riemann's  surface  associated 
with  a  multiform  function  is  generally  one  of  multiple  connection  ;  we  shall 
find  it  convenient  to  know  how  it  can  be  resolved,  by  means  of  cross-cuts,  into 
a  simply  connected  surface.  The  representative  surface  will  be  supposed  a 
closed  surface  with  a  single  boundary  ;  its  connectivity,  necessarily  odd,  being 
2/)  +  l,  the  number  of  cross-cuts  necessary  to  resolve  the  surface  into  one 
that  is  simply  connected  is  2p  ;  when  these  cuts  have  been  made,  the  simply 
connected  surface  then  obtained  will  have  its  boundary  composed  of  a  single 
closed  curve. 


179.] 


BY    CROSS-CUTS 


351 


One  or  two  simple  examples  of  resolution  of  special  Riemann's  surfaces  will  be  useful 
in  leading  up  to  the  general  explanation  ;  in  the  examples  it  will  be  shewn  how,  in 
conformity  with  §  168,  the  resolving  cross-cuts  render  irreducible  circuits  impossible. 

Ex.  1.     Let  *he  equation  be 

w1  =  A(z-d)(z-b'](z-c}(z-d\ 

where  a,  b,  c,  d  are  four  distinct  points,  all  of  finite  modulus.  The  surface  is  two-sheeted ; 
each  of  the  points  a,  b,  c,  d  is  a  branch-point  where  the  two  values  of  w  interchange  ;  and 
so  the  surface,  assumed  to  have  a  single  boundary,  is  triply  connected,  the  value  of  p 
being  unity.  The  branch-lines  are  two,  each  connecting  a  pair  of  branch-points  ;  let  them 
be  ab  and  cd. 

Two  cross-cuts  are  necessary  and  sufficient  to  resolve  the  surface  into  one  that  is 
simply  connected.  We  first  make  a  cross-cut, 
beginning  at  the  boundary  S,  (say  it  is  in  the 
upper  sheet),  continuing  in  that  sheet  and  re 
turning  to  J3,  so  that  its  course  encloses  the 
branch-line  ab  (but  not  cd)  and  meets  no  branch- 
line.  It  is  a  cross-cut,  and  not  a  loop-cut,  for  it 
begins  and  ends  in  the  boundary  ;  it  is  evidently 
a  cut  in  the  upper  sheet  alone,  and  does  not 
divide  the  surface  into  distinct  portions  ;  and, 
once  made,  it  is  to  be  regarded  as  boundary  for 
the  partially  cut  surface. 

The  surface  in  its  present  condition  is  con 
nected  :  and  therefore  it  is  possible  to  pass  from  one  edge  to  the  other  of  the  cut  just 
made.  Let  P  be  a  point  on  it  ;  a  curve  that  passes  from  one  edge  to  the  other  is  indicated 
by  the  line  PQR  in  the  upper  sheet,  RS  in  the  lower,  and  SP  in  the  upper.  Along  this 
line  make  a  cut,  beginning  at  P  and  returning  to  P  ;  it  is  a  cross-cut,  partly  in  the 
upper  sheet  and  partly  in  the  lower,  and  it  does  not  divide  the  surface  into  distinct 
portions. 

Two  cross-cuts  in  the  triply  connected  surface  have  now  been  made  ;  neither  of  them, 
as  made,  divides  the  surface  into  distinct  portions,  and  each  of  them  when  made  reduces 
the  connectivity  by  one  unit ;  hence  the  surface  is  now  simply  connected.  It  is  easy  to 
see  that  the  boundary  consists  of  a  single  line  not  intersecting  itself;  for  beginning 
at  P,  we  have  the  outer  edge  of  PUT,  then  the  inner  edge  of  2'QltSP,  then  the  inner 
edge  of  PTB,  and  then  the  outer  edge  of  PSRQP,  returning  to  P. 

The  required  resolution  has  been  effected. 

Before  the  surface  was  resolved,  a  number  of  irreducible  circuits  could  be  drawn  ;  a 
complete  system  of  irreducible  circuits  is  composed  of  two,  by  §  168.  Such  a  system  may 
be  taken  in  various  ways  ;  let  it  be  composed  of  a  simple  curve  C  lying  in  the  upper  sheet 
and  containing  the  points  a  and  b,  and  a  simple  curve  D,  lying  partly  in  the  upper 
and  partly  in  the  lower  sheet  and  containing  the  points  a  and  c  ;  each  of  these  curves 
is  irreducible,  because  it  encloses  two  branch-points.  Every  other  irreducible  circuit 
is  reconcileable  with  these  two  ;  the  actual  reconciliation  in  particular  cases  is  effected 
most  simply  when  the  surface  is  taken  in  a  spherical  form. 

The  irreducible  circuit  C  on  the  unresolved  surface  is  impossible  on  the  resolved 
surface  owing  to  the  cross-cut  SPQRS ;  and  the  irreducible  circuit  D  on  the  unresolved 
surface  is  impossible  on  the  resolved  surface  owing  to  the  cross-cut  PTB.  It  is  easy 
to  verify  that  no  irreducible  circuit  can  be  drawn  on  the  resolved  surface. 


352 


RESOLUTION 


[179. 


In  practice,  it  is  conveniently  effective  to  select  a  complete  system  of  irreducible 
simple  circuits  and  then  to  make  the  cross-cuts  so  that  each  of  them  renders  one  circuit 
of  the  system  impossible  on  the  resolved  surface. 

Ex.  2.     If  the  equation  be 


=  4:(z-e1)(z-e.t)(z-e3), 

the  branch-points  are  els  e2,  e3  and  oo  .  When  the  two-sheeted  surface  is  spherical,  and  the 
branch-lines  are  taken  to  be  (i)  a  line  joining  elf  e.2',  and  (ii)  a  line  joining  e3  to  the  South 
pole,  the  discussion  of  the  surface  is  similar  in  detail  to  that  in  the  preceding  example. 

Ex.  3.     Let  the  equation  be 

t*«-4*  (!-«)(*-*)  <X-*)&*-«), 

and  for  simplicity  suppose  that  AC,  X,  /*  are  real  quantities  subject  to  the  inequalities 


The  associated  surface  is  two-sheeted  and  has  a  boundary  assigned  to  it  ;  assuming 
that  its  sheets  are  planes,  we  shall  take  some  point  in  the  finite  part  of  the  upper  sheet, 
not  being  a  branch-point,  as  the  boundary.  There  are  six  .  branch-points,  viz.,  0,  1,  K, 
X,  /x,  co  at  each  of  which  the  two  values  of  w  interchange  ;  and  so  the  connectivity  of  the 
surface  is  5  and  its  class,  p,  is  2.  The  branch-lines  can  be  taken  as  three,  this  being 
the  simplest  arrangement  ;  let  them  be  the  lines  joining  0,  1  ;  K,  X  ;  /*,  oo  . 

Four  cross-cuts  are  necessary  to  resolve  the  surface  into  one  that  is  simply  connected 
and  has  a  single  boundary.  They  may  be  obtained  as  follows. 


Fig.  62. 

Beginning  at  the  boundary  L,  let  a  cut  LHA  be  made  entirely  in  the  upper  sheet 
along  a  line  which,  when  complete,  encloses  the  points  0  and  1  but  no  other  branch-points ; 
let  the  cut  return  to  L.  This  is  a  cross-cut  and  it  does  not  divide  the  surface  into 
distinct  pieces ;  hence,  after  it  is  made,  the  connectivity  of  the  modified  surface  is  4,  and 
there  are  two  boundary  lines,  being  the  two  edges  of  the  cut  LHA. 

Beginning  at  a  point  A  in  LHA,  make  a  cut  along  ABC  in  the  upper  sheet  until 
it  meets  the  branch-line  /zoo,  then  in  the  lower  sheet  along  CSD  until  it  meets  the 
branch-line  01,  and  then  in  the  upper  sheet  from  D  returning  to  the  initial  point  A. 
This  is  a  cross-cut  and  it  does  not  divide  the  surface  into  distinct  pieces ;  hence,  after  it 
is  made,  the  connectivity  of  the  modified  surface  is  3,  and  it  is  easy  to  see  that  there 
is  only  one  boundary  edge,  similar  to  the  single  boundary  in  Ex.  1  when  the  surface 
in  that  example  has  been  completely  resolved. 

Make  a  loop-cut  EFG  along  a  line,  enclosing  the  points  K  and  X  but  no  other  branch 
points  ;  and  change  it  into  a  cross-cut  by  making  a  cut  from  E  to  some  point  B  of  the 
boundary.  This  cross-cut  can  be  regarded  as  BEFGE,  ending  at  a  point  in  its  own 
earlier  course.  As  it  does  not  divide  the  surface  into  distinct  pieces,  the  connectivity  is 
reduced  to  2  ;  and  there  are  two  boundary  lines. 


179.]  BY   CROSS-CUTS  353 

Beginning  at  a  point  G  make  another  cross-cut  GQPRG,  as  in  the  figure,  enclosing 
the  two  branch-points  X  and  p,  and  lying  partly  in  the  upper  sheet  and  partly  in  the  lower. 
It  does  not  divide  the  surface  into  distinct  pieces  :  the  connectivity  is  reduced  to  unity 
and  there  is  a  single  boundary  line. 

Four  cross-cuts  have  been  made  ;  and  the  surface  has  been  resolved  into  one  that  is 
simply  connected. 

It  is  easy  to  verify  : 

(i)  that  neither  in  the  upper  sheet,  nor  in  the  lower  sheet,  nor  partly  in  the 
upper  sheet  and  partly  in  the  lower,  can  an  irreducible  circuit  be  drawn  in  the  resolved 
surface  ;  and 

•  (ii)  that,  owing  to  the  cross-cuts,  the  simplest  irreducible  circuits  in  the  unresolved 
surface — viz.  those  which  enclose  0,  1  ;  1,  K  ;  *,  X  ;  X,  /i  ;  respectively — are  rendered 
impossible  in  the  resolved  surface. 

The  equation  in  the  present  example,  and  the  Riemann's  surface  associated  with  it, 
lead  to  the  theory  of  hyperelliptic  functions*. 

180.  The  last  example  suggests  a  method  of  resolving  any  two-sheeted 
surface  into  a  surface  that  is  simply  connected. 

The  number  of  its  branch-points  is  necessarily  even,  say  2p  +  2.  The 
branch-lines  can  be  made  to  join  these  points  in  pairs,  so  that  there  will  be 
p  +  l  of  them.  To  determine  the  connectivity  (§  178),  we  have  n  =  2  and, 
since  two  values  are  interchanged  at  every  branch-point,  H  =  2p  -f  2 ;  so 
that  the  connectivity  is  2p  + 1.  Then  2p  cross-cuts  are  necessary  for  the 
required  resolution  of  the  surface. 

We  make  cuts  round  p  of  the  branch-lines,  that  is,  round  all  of  them  but 
one ;  each  cut  is  made  to  enclose  two  branch-points,  and  each  lies  entirely  in 
the  upper  sheet.  These  are  cuts  corresponding  to  the  cuts  LHA  and  EFG 
in  fig.  62 ;  and,  as  there,  the  cut  round  the  first  branch-line  begins  and  ends 
in  the  boundary,  so  that  it  is  a  cross-cut.  All  the  remaining  cuts  are  loop- 
cuts  at  present.  The  system  of  p  cuts  we  denote  by  a1}  a2,  ...,  ap. 

We  make  other  p  cuts,  one  passing  from  the  inner  edge  of  each  of  the  p 
cuts  a  already  made  to  the  branch-line  which  it  surrounds,  then  in  the  lower 
sheet  to  the  (j)  +  l)th  branch-line,  and  then  in  the  upper  sheet  returning  to 
the  point  of  the  outer  edge  of  the  cut  a  at  which  it  began.  This  system  of 
cuts  corresponds  to  the  cuts  ADSGBA  and  GQPRG  in  fig.  62.  Each  of  them 
can  be  taken  so  as  to  meet  no  one  of  the  cuts  a  except  the  one  in  which  it 
begins  and  ends ;  and  they  can  be  taken  so  as  not  to  meet  one  another. 
This  system  of  p  cuts  we  denote  by  bl}  b.2,  ...,  bp,  where  br  is  the  cut  which 
begins  and  ends  in  ar.  All  these  cuts  are  cross-cuts,  because  they  begin  and 
end  in  boundary-lines. 

Lastly,  we  make  other  p  —  1  cuts  from  ar  to  6.r_1}  for  r  =  2,  3,  . ..,  p,  all  in 

*  One  of  the  most  direct  discussions  of  the  theory  from  this  point  of  view  is  given  by  Prym, 
Neue  Theorie  der  ultraelliptischen  Functionen,  (Berlin,  Mayer  and  Miiller,  2nd  ed.,  1885). 

F.  23 


354  GENERAL   RESOLUTION   OF   SURFACE  [180. 

the  upper  sheet ;  no  one  of  them,  except  at  its  initial  and  its  final  points, 
meets  any  of  the  cuts  already  made.  This  system  of  p  -  1  cuts  we  denote 
by  c% ,  GS,  . . . ,  Cp . 

Because  br^  is  a  cross-cut,  the  cross-cut  cr  changes  ar  (hitherto  a  loop- 
cut)  into  a  cross-cut  when  cr  and  ar  are  combined  into  a  single  cut. 

It  is  evident  that  no  one  of  these  cuts  divides  the  surface  into  distinct 
pieces;  and  thus  we  have  a  system  of  2p  cross-cuts  resolving  the  two-sheeted 
surface  of  connectivity  2p+I  into  a  surface  that  is  simply  connected.  The 
cross-cuts  in  order*  are 

Oj,  &j,  C2  and  aa,  62,  c3  and  as,  bs,  ...,cp  and  ap,  bp. 

181.     This  resolution  of  a  general  two-sheeted  surface  suggests  f  Rie- 

mann's  general  resolution  of  a  surface  with  any  (finite)  number  of  sheets. 

As  before,  we  assume  that  the  surface  is  closed  and  has  a  single  boundary 

and  that  its  class  is  p,  so  that  2p  cross-cuts  are  necessary  for  its  resolution 

into  one  that  is  simply  connected. 

Make  a  cut  in  the  surface  such  as  not  to  divide  it  into  distinct  pieces; 
and  let  it  begin  and  end  in  the  boundary.  It  is  a  cross-cut,  say  ^ ;  it 
changes  the  number  of  boundary-lines  to  2  and  it  reduces  the  connectivity 
of  the  cut  surface  to  2p. 

Since  the  surface  is  connected,  we  can  pass  in  the  surface  along  a 
continuous  line  from  one  edge  of  the  cut  ^  to  the  opposite  edge.  Along 
this  line  make  a  cut  6j :  it  is  a  cross-cut,  because  it  begins  and  ends  in 
the  boundary.  It  passes  from  one  edge  of  c^  to  the  other,  that  is,  from  one 
boundary-line  to  another.  Hence,  as  in  Prop.  II.  of  §  164,  it  does  not  divide  ' 
the  surface  into  distinct  pieces;  it  changes  the  number  of  boundaries  to  1 
and  it  reduces  the  connectivity  to  1p  —  1. 

The  problem  is  now  the  same  as  at  first,  except  that  now  only 
2«  —  2  cross-cuts  are  necessary  for  the  required  resolution.  We  make  a 
loop-cut  a.2,  not  resolving  the  surface  into  distinct  pieces,  and  a  cross-cut 
d  from  a  point  of  a2  to  a  point  on  the  boundary  at  6j ;  then  Cj  and  a,2>  taken 
together,  constitute  a  cross-cut  that  does  not  resolve  the  surface  into  distinct 
pieces.  It  therefore  reduces  the  connectivity  to  2p  —  2  and  leaves  two  pieces 
of  boundary. 

The  surface  being  connected,  we  can  pass  in  the  surface  along  a  continuous 
line  from  one  edge  of  a»  to  the  opposite  edge.  Along  this  line  we  make  a  cut 
b.2,  evidently  a  cross-cut,  passing,  like  h  in  the  earlier  case,  from  one 
boundary-line  to  the  other.  Hence  it  does  not  divide  the  surface  into 

*  See  Neumann,  pp.  178 — 182;  Prym,  Zur  Thcorie  der  Fwwtionen  in  einer  zweiblattrigen 
Flfahe,  (1866). 

+  Riemann,  Ges.  Werke,  pp.  122,  123 ;  Neumann,  pp.  182—185. 


181.]  BY  CROSS-CUTS  355 

distinct  pieces;  it  changes  the  number  of  boundaries  to  1  and  it  reduces 
the  connectivity  to  2p  —  3. 

Proceeding  in  p  stages,  each  of  two  cross-cuts,  we  ultimately  obtain  a 
simply  connected  surface  with  a  single  boundary ;  and  the  general  effect  on 
the  original  unresolved  surface  is  to  have  a  system  of  cross-cuts  somewhat  of 
the  form 


Fig.  63. 

The  foregoing  resolution  is  called  the  canonical  resolution  of  a  Riemann's 
surface. 

Ex.  1.     Construct  the  Riemann's  surface  for  the  equation 

w3  +  z3  —  3awz—  1, 

both  for  a  =  0  and  for  a  different  from  zero;  and  resolve  it  by  cross-cuts  into  a  simply 
connected  surface  with  a  single  boundary,  shewing  a  complete  system  of  irreducible  simple 
circuits  on  the  unresolved  surface. 

Ex.  2.     Shew  that  the  Riemann's  surface  for  the  equation 


_ 
(z-c)(z-d) 

is  of  class  p  =  2-  indicate  the  possible  systems  of  branch-lines,  and,  for  each  system, 
resolve  the  surface  by  cross-cuts  into  a  simply  connected  surface  with  a  single  boundary. 

(Burnside.) 

182.  Among  algebraical  equations  with  their  associated  Riemann's 
surfaces,  two  general  cases  of  great  importance  and  comparative  simplicity 
distinguish  themselves..  The  first  is  that  in  which  the  surface  is  two- 
sheeted  ;  round  each  branch-point  the  two  branches  interchange.  The 
second  is  that  in  which,  while  the  surface  has  a  finite  number  of  sheets 
greater  than  two,  all  the  branch-points  are  of  the  first  order,  that  is,  are 
such  that  round  each  of  them  only  two  branches  of  the  function  interchange. 
The  former  has  already  been  considered,  in  so  for  as  concerns  the  surface  ; 
we  now  proceed  to  the  consideration  of  the  latter. 

The  equation  is  f(w,  z)  =  0, 

of  degree  n  in  w;   and,  for  our  present  purpose,  it  is  convenient  to  regard 

0  as  an  equation  corresponding  to  a  generalised  plane  curve  of  degree  n 
so  that  no  term  in  /  is  of  dimensions  higher  than  n. 

The  total  number  of  branch-points  has  been  proved,  in  §  98,  to  be 

w(w-l)-28-2«, 

23—2 


356  DEFICIENCY  [182. 

where  S  is  the  number  of  points  which  are  the  generalisation  of  double 
points  on  the  curve  with  non-coincident  tangents  and  K  is  the  number 
of  double  points  on  the  curve  with  coincident  tangents.  Round  each  of 
these  branch-points,  two  branches  of  w  interchange  and  only  two,  so  that 
all  the  numbers  mq  of  §  178  are  equal  to  2  ;  hence  the  ramification 
H  is 

2  [n  (n  -  1)  -  2S  -  2/e}  -  [n  (n  -  1)  -  2S  -  2*}, 

that  is,  n=w(n-l)-28-2«. 

The  connectivity  of  the  surface  is  therefore 

w  (n  -  1)  -  28  -  2*  -  2n  +  3  ; 
and  therefore  the  class  p  of  the  surface  is 

£(n-l)(»-2)-8-«. 

Now  this  integer  is  known*  as  the  deficiency  of  the  curve;  and  therefore  it 
appears  that  the  deficiency  of  the  curve  is  the  same  as  the  class  of  the  Riemann 
surface  associated  with  its  equation,  and  also  is  the  same  as  the  class  of  its 
equation. 

Moreover,  the  number  of  branch-points  of  the  original  equation  is  fl,  that 

is, 

n  -  2 


Note.  The  equality  of  these  numbers,  representing  the  deficiency  and 
the  class,  is  one  among  many  reasons  that  lead  to  the  close  association  of 
algebraic  functions  (and  of  functions  dependent  on  them)  with  the  theory  of 
plane  algebraic  curves,  in  the  investigations  of  Nb'ther,  Brill,  Clebsch  and 
others,  referred  to  in  §§  191,  242. 

183.  With  a  view  to  the  construction  of  a  canonical  form  of  Riemann's 
surface  of  class  p  for  the  equation  under  consideration,  it  is  necessary  to 
consider  in  some  detail  the  relations  between  the  branches  of  the  functions 
as  they  are  affected  by  the  branch-  points. 

The  effect  produced  on  any  value  of  the  function  by  the  description  of  a 
small  circuit,  enclosing  one  branch-point  (and  only  one),  is  known.  But 
when  the  small  circuit  is  part  of  a  loop,  the  effect  on  the  value  of  the 
function  with  which  the  loop  begins  to  be  described  depends  upon  the  form 
of  the  loop;  and  various  results  (e.g.  Ex.  1,  §  104)  are  obtained  by  taking 
different  loops.  In  the  first  form  (§  175)  in  which  the  branch-lines  were 
established  as  junctions  between  sheets,  what  was  done  was  the  equivalent 

*  Salmon's  Higher  Plane  Curves,  §§  44,  83;  Clebsch's  Vorlesungen  iiber  Geometrie,  (edited 
by  Lindemann),  t.  i,  pp.  351  —  429,  the  German  word  used  instead  of  deficiency  being  Geschlecht. 
The  name  'deficiency'  was  introduced  by  Cayley  in  1865:  see  Proc.  Land.  Math.  Soc.,  vol.  i., 
"  On  the  transformation  of  plane  curves." 


183.]  LOOPS  357 

of  drawing  a  number  of  straight  loops,  which  had  one  extremity  common  to 
all  and  the  other  free,  and  of  assigning  the  law  of  junction  according  to  the 
law  of  interchange  determined  by  the  description  of  the  loop.  As,  however, 
there  is  no  necessary  limitation  to  the  forms  of  branch-lines,  we  may  draw 
them  in  other  forms,  always,  of  course,  having  branch-points  at  their  free 
extremities ;  and  according  to  the  variation  in  the  form  of  the  branch-line, 
(that  is,  according  to  the  variation  in  the  form  of  the  corresponding  loop 
or,  in  other  words,  according  to  the  deformation  of  the  loop  over  other 
branch-points  from  some  form  of  reference),  there  will  be  variation  in  the  law 
of  junction  along  the  branch-lines. 

There  is  thus  a  large  amount  of  arbitrary  character  in  the  forms  of  the 
branch-lines,  and  consequently  in  the  laws  of  junction  along  the  branch-lines, 
of  the  sheets  of  a  Riemann's  surface.  Moreover,  the  assignment  of  the  n 
branches  of  the  function  to  the  n  sheets  is  arbitrary.  Hence  a  consider 
able  amount  of  arbitrary  variation  in  the  configuration  of  a  Riemann's 
surface  is  possible  within  the  limits  imposed  by  the  invariance  of  its 
connectivity.  The  canonical  form  will  be  established  by  making  these 
arbitrary  elements  definite. 

184.  After  the  preceding  explanation  and  always  under  the  hypothesis 
that  the  branch-points  are  simple,  we  shall  revert  temporarily  to  the  use  of 
loops  and  shall  ultimately  combine  them  into  branch-lines. 

When,  with  an  ordinary  point  as  origin,  we  construct  a  loop  round  a 
branch-point,  two  and  only  two  of  the  values  of  the  function  are  affected 
by  that  particular  loop ;  they  are  interchanged  by  it ;  but  a  different  form  of 
loop,  from  the  same  origin  round  the  same  branch-point,  might  affect  some 
other  pair  of  values  of  the  function. 

To  indicate  the  law  of  interchange,  a  symbol  will  be  convenient.  If  the 
two  values  interchanged  by  a  given  loop  be  Wi  and  wm,  the  loop  will  be 
denoted  by  im  ;  and  i  and  ra  will  be  called  the  numbers  of  the  symbol  of  that 
loop. 

For  the  initial  configuration  of  the  loops,  we  shall  (as  in  §  175)  take  an 
ordinary  point  0 :  we  shall  make  loops  beginning  at  0,  forming  them  in  the 
sequence  of  angular  succession  of  the  branch-points  round  0  and  drawing  the 
double  linear  part  of  the  loop  as  direct  as  possible  from  0  to  its  branch-point : 
and,  in  this  configuration,  we  shall  take  the  law  of  interchange  by  a  loop  to 
be  the  law  of  interchange  by  the  branch-point  in  the  loop. 

In  any  other  configuration,  the  symbol  of  a  loop  round  any  branch-point 
depends  upon  its  form,  that  is,  depends  upon  the  deformation  over  other 
branch-points  which  the  loop  has  suffered  in  passing  from  its  initial  form. 
The  effect  of  such  deformation  must  first  be  obtained :  it  is  determined  by 
the  following  lemma : — 


358 


MODIFICATION 


[184. 


When  one  loop  is  deformed  over  another,  the  symbol  of  the  deformed  loop  is 
unaltered,  if  neither  of  its  numbers  or  if  both  of  its  numbers  occur  in  the 
symbol  of  the  u